Differentiation | Practice Problems, Examples & Solutions

Derivatives of Polynomials and Exponential Functions

21st Century Astronomy

When viewed by radio telescopes, Jupiter is the second-brightest object in the sky. What is the source of its radiation?

Worlds of Gas and Liquid-The Giant Planets

01:36

Calculus for Scientists and Engineers: Early Transcendental

Calculating limits exactly Use the definition of the derivative to evaluate the following limits.
$$\lim _{h \rightarrow 0} \frac{(3+h)^{3+h}-27}{h}$$

Derivatives

Derivatives of Logarithmic and Exponential Functions

00:57

Calculus for Scientists and Engineers: Early Transcendental

Graph the functions $f(x)=x^{3}, g(x)=3^{x}$ and $h(x)=x^{x}$ and find their common intersection point (exactly).

Derivatives

Derivatives of Logarithmic and Exponential Functions

Product and Quotient Rules

225 Practice Problems

02:22

Calculus: Early Transcendental Functions

Find an antiderivative by reversing the chain rule, product rule or quotient rule.
$$\int\left(x \sin 2 x+x^{2} \cos 2 x\right) d x$$

Integration

Antiderivatives

01:50

Calculus: Early Transcendental Functions

Use the product rule to show that if $g(x)=[f(x)]^{2}$ and $f(x)$ is differentiated, then $g^{\prime}(x)=2 f(x) f^{\prime}(x) .$ This is an example of the chain rule, to be discussed in section 2.5

Differentiation

The Product and Quotient Rules

03:16

Calculus: Early Transcendental Functions

Find the derivative of each function.
$$f(x)=\frac{x^{2}+2 x+5}{x^{2}-5 x+1}$$

Differentiation

The Product and Quotient Rules

Derivatives of Trigonometric Functions

188 Practice Problems

00:56

Calculus for Scientists and Engineers: Early Transcendental

Identifying derivatives from limits The following limits equal the derivative of a function $f$ at a point a.
a. Find one possible $f$ and $a$
b. Evaluate the limit.
$$\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{6}+h\right)-\frac{1}{2}}{h}$$

Derivatives

Derivatives of Trigonometric Functions

02:36

Calculus for Scientists and Engineers: Early Transcendental

Continuity of a piecewise function Let $$f(x)=\left\{\begin{array}{cc}\frac{3 \sin x}{x} & \text { if } x \neq 0 \\a & \text { if } x=0\end{array}\right.$$ For what values of $a$ is $f$ continuous?

Derivatives

Derivatives of Trigonometric Functions

02:34

Calculus for Scientists and Engineers: Early Transcendental

Proof of $\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0$ Use the trigonometric identity $\cos ^{2} x+\sin ^{2} x=1$ to prove that $\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 .$ (Hint: Begin by multiplying the numerator and denominator by $\cos x+1 .)$

Derivatives

Derivatives of Trigonometric Functions

The Chain Rule

366 Practice Problems

12:35

Calculus for Scientists and Engineers: Early Transcendental

Let $f(x, y)=0$ define $y$ as a twice differentiable function of $x$
a. Show that $y^{\prime \prime}(x)=-\frac{f_{x x} f_{y}^{2}-2 f_{x} f_{y} f_{x y}+f_{y y} f_{x}^{2}}{f_{y}^{3}}$.
b. Verify part (a) using the function $f(x, y)=x y-1$.

Functions of Several Variables

The Chain Rule

02:03

Calculus for Scientists and Engineers: Early Transcendental

Consider the following surfaces specified in the form $z=f(x, y)$ and the curve $C$ in the $x y$ -plane given parametrically in the form $x=g(t), y=h(t)$.
a. In each case, find $z^{\prime}(t)$.
b. Imagine that you are walking on the surface directly above the curve $C$ in the direction of increasing t. Find the values of t for which you are walking uphill (that is, z is increasing).
$$z=2 x^{2}+y^{2}+1, C: x=1+\cos t, y=\sin t ; 0 \leq t \leq 2 \pi$$

Functions of Several Variables

The Chain Rule

06:50

Calculus for Scientists and Engineers: Early Transcendental

Assume that $F(x, y, z(x, y))=0$ implicitly defines $z$ as a differentiable function of $x$ and $y .$ Extend Theorem 13.9 to show that
$$\frac{\partial z}{\partial x}=-\frac{F_{x}}{F_{z}} \text { and } \frac{\partial z}{\partial y}=-\frac{F_{y}}{F_{z}}$$

Functions of Several Variables

The Chain Rule

Implicit Differentiation

200 Practice Problems

06:59

Calculus for Scientists and Engineers: Early Transcendental

A challenging derivative Find $\frac{d y}{d x},$ where $\sqrt{3 x^{7}+y^{2}}=\sin ^{2} y+100 x y$

Derivatives

Implicit Differentiation

01:46

Calculus for Scientists and Engineers: Early Transcendental

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals. . A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas $y=c x^{2}$ form orthogonal trajectories with the family of ellipses $x^{2}+2 y^{2}=k,$ where $c$ and $k$ are constants (see figure). Use implicit differentiation if needed to find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. CANT COPY THE GRAPH. $y=m x ; x^{2}+y^{2}=a^{2},$ where $m$ and $a$ are constants

Derivatives

Implicit Differentiation

01:33

Calculus for Scientists and Engineers: Early Transcendental

Normal lines $A$ normal line on a curve passes through a point P on the curve perpendicular to the line tangent to the curve at $P(\text {see figure}) .$ Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. CANT COPY THE GRAPH Exercise 28

Derivatives

Implicit Differentiation

Derivatives of Logarithmic Functions

154 Practice Problems

02:11

Calculus: Early Transcendentals

Logarithmic differentiation Use logarithmic differentiation to evaluate $f^{\prime}(x)$.
$$f(x)=x^{10 x}$$

Derivatives

Derivatives of Logarithmic and Exponential Functions

00:52

Calculus: Early Transcendentals

Find the derivative of the following functions.
$$y=\ln \left|x^{2}-1\right|$$

Derivatives

Derivatives of Logarithmic and Exponential Functions

00:45

Calculus: Early Transcendentals

Find the derivative of the following functions.
$$y=\ln 2 x^{8}$$

Derivatives

Derivatives of Logarithmic and Exponential Functions

Exponential Growth and Decay

280 Practice Problems

01:43

Introductory and Intermediate Algebra for College Students

Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
In $2006,$ Canada's population exceeded Uganda's by
4.9 million.

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

01:41

Introductory and Intermediate Algebra for College Students

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning.
After 100 years, a population whose growth rate is $3 \%$ will have three times as many people as a population whose growth rate is $1 \%$

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

04:03

Introductory and Intermediate Algebra for College Students

We used two data points and an exponential function to model the population of the United States from 1970 through 2009. The data are shown again in the table. Use all five data points.
$$\begin{array}{c|c}
\hline x, \text { Number of Years after } 1969 & y, \text { U.S. Population (millions) } \\
\hline 1(1970) & 203.3 \\
\hline 11(1980) & 226.5 \\
\hline 21(1990) & 248.7 \\
\hline 31(2000) & 281.4 \\
\hline 40(2009) & 307.0 \\
\hline
\end{array}$$
Use the values of $r$ in Exercises $45-48$ to select the two models of best fit. Use each of these models to predict by which year the U.S. population will reach 352 million. How do these answers compare to the year we found in Example $1,$ namely $2020 ?$ If you obtained different years, how do you account for this difference?

Exponential and Logarithmic Functions

Exponential Growth and Decay; Modeling Data

Related Rates

156 Practice Problems

01:05

Organic Chemistry

Draw a reaction coordinate diagram for the following reaction in which $C$ is the most stable and $B$ the least stable of the three species and the transition state going from $A$ to $B$ is more stable than the transition state going from B to C:
$$A \stackrel{k_{1}}{E_{-1}} \quad B \quad \frac{k_{2}}{\overline{k_{-2}}} C$$
a. How many intermediates are there?
b. How many transition states are there?
c. Which step has the greater rate constant in the forward direction?
d. Which step has the greater rate constant in the reverse direction?
e. Of the four steps, which has the greatest rate constant?
f. Which is the rate-determining step in the forward direction?
g. Which is the rate-determining step in the reverse direction?

Alkenes: Structure, Nomenclature, and an Introduction to Reactivity • Thermodynamics and Kinetics

01:15

University Physics

How does the self-inductance per unit length near the center of a solenoid (away from the ends) compare with its value near the end of the solenoid?

Inductance

02:21

Physics

The Sun emits electromagnetic waves (including light) equally in all directions. The intensity of the waves at Earth's upper atmosphere is $1.4 \mathrm{kW} / \mathrm{m}^{2} .$ At what rate does the Sun emit electromagnetic waves? (In other words, what is the power output?)

Waves

Linear Approximation and Differentials

149 Practice Problems

00:39

Calculus for Scientists and Engineers: Early Transcendental

Differentials Consider the following functions and express the relationship between a small change in $x$ and the corresponding change in $y$ in the form $d y=f^{\prime}(x) d x$.
$$f(x)=3 x^{3}-4 x$$

Applications of the Derivative

Linear Approximation and Differentials

01:30

Calculus for Scientists and Engineers: Early Transcendental

Approximate the change in the magnitude of the electrostatic force between two charges when the distance between them increases from $r=20 \mathrm{m}$ to $r=21 \mathrm{m}\left(F(r)=0.01 / r^{2}\right)$.

Applications of the Derivative

Linear Approximation and Differentials

02:25

Calculus for Scientists and Engineers: Early Transcendental

Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
$$1 / \sqrt[3]{510}$$

Applications of the Derivative

Linear Approximation and Differentials

Hyperbolic Functions

173 Practice Problems

0:00

Mathematical Statistics with Applications

Applet Exercise In Exercise 16.15, we determined that the posterior density for $p$, the proportion of responders to the new treatment for a virulent disease, is a beta density with parameters $\alpha^{*}=5$ and $\beta^{*}=24 .$ What is the conclusion of a Bayesian test for $H_{0}: p<.3$ versus $H_{a}:$ $p \geq .3 ?$ [Use the applet Beta Probabilities and Quantiles at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html.
Alternatively, if $W$ is a beta-distributed random variable with parameters $\alpha$ and $\beta$, the $R$ or $S$ -Plus command pbeta $(w, \alpha, \beta) \text { gives } P(W \leq w) .]$

Introduction to Bayesian Methods for Inference

Bayesian Tests of Hypotheses

02:12

Calculus for Scientists and Engineers: Early Transcendental

Carry out the following steps to derive the formula $\left.\int \operatorname{csch} x \, d x=\ln |\tanh (x / 2)|+C \text { (Theorem } 6.9\right)$
a. Change variables with the substitution $u=x / 2$ to show that
$$\int \operatorname{csch} x \, d x=\int \frac{2 d u}{\sinh 2 u}$$
b. Use the identity for sinh $2 u$ to show that $\frac{2}{\sinh 2 u}=\frac{\operatorname{sech}^{2} u}{\tanh u}$
c. Change variables again to determine $\int \frac{\operatorname{sech}^{2} u}{\tanh u} d u,$ and then express your answer in terms of $x$

Applications of Integration

Hyperbolic Functions

01:03

Calculus for Scientists and Engineers: Early Transcendental

Verify the following identities.
$$\sinh \left(\cosh ^{-1} x\right)=\sqrt{x^{2}-1}, \text { for } x \geq 1$$

Applications of Integration

Hyperbolic Functions

Maximum and Minimum Values

540 Practice Problems

02:36

Calculus for Scientists and Engineers: Early Transcendental

a. Find the critical points of the following functions on the given interval.
b. Use a graphing device to determine whether the critical points correspond to local maxima, local minima, or neither.
c. Find the absolute maximum and minimum values on the given interval when they exist.
$$f(x)=x^{2 / 3}\left(4-x^{2}\right) ;[-3,4]$$

Applications of the Derivative

Maxima and Minima

02:27

Calculus for Scientists and Engineers: Early Transcendental

a. Find the critical points of $f$ on the given interval.
b. Determine the absolute extreme values of $f$ on the given interval.
c. Use a graphing utility to confirm your conclusions.
$$f(x)=x \ln (x / 5) ;[0.1,5]$$

Applications of the Derivative

Maxima and Minima

01:30

Calculus for Scientists and Engineers: Early Transcendental

All rectangles with an area of 64 have a perimeter given by $P(x)=2 x+128 / x,$ where $x$ is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function. What are the dimensions of the rectangle with minimum perimeter?

Applications of the Derivative

Maxima and Minima

The Mean Value Theorem

82 Practice Problems

04:49

Statistics Informed Decisions Using Data

What is the mean square due to treatment estimate of $\sigma^{2} ?$ What is the mean square due to error estimate of $\sigma^{2} ?$

Comparing Three or More Means

01:34

Calculus for Scientists and Engineers: Early Transcendental

Running pace Explain why if a runner completes a 6.2 -mi $(10-\mathrm{km})$ race in 32 min, then he must have been running at exactly $11 \mathrm{mi} / \mathrm{hr}$ at least twice in the race. Assume the runner's speed at the finish line is zero.

Applications of the Derivative

Mean Value Theorem

02:16

Calculus for Scientists and Engineers: Early Transcendental

Mean Value Theorem and graphs By visual inspection, locate all points on the graph at which the slope of the tangent line equals the average rate of change of the function on the interval [-4,4]
(GRAPH CAN'T COPY)

Applications of the Derivative

Mean Value Theorem

How Derivatives Affect the Shape of a Graph

343 Practice Problems

00:43

Calculus: Early Transcendental Functions

Estimate the intervals where the function is concave up and concave down. (Hint: Estimate where the slope is increasing and decreasing.)
(GRAPH CAN'T COPY)

Applications of Differentiation

Concavity and The Second Derivative Test

03:20

Calculus: Early Transcendental Functions

The "family of functions" contains a parameter $c .$ The value of $c$ affects the properties of the functions. Determine what differences, if any, there are for $c$ being zero, positive or negative. Then determine what the graph would look like for very large positive $c$ 's and for very large negative $c$ 's.
$$f(x)=x^{4}+c x^{2}$$

Applications of Differentiation

Overview of Curve Sketching

01:18

Calculus: Early Transcendental Functions

Determine all significant features (approximately if necessary) and sketch a graph.
$$f(x)=\sin x-\frac{1}{2} \sin 2 x$$

Applications of Differentiation

Overview of Curve Sketching

Indeterminate Forms

35 Practice Problems

02:15

Calculus: Early Transcendental Functions

Find the indicated limits.
$$\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}$$

Applications of Differentiation

Indeterminate Forms and L'Hopitals Rule

04:16

Calculus: Early Transcendental Functions

Find the indicated limits.
$$\lim _{x \rightarrow 0} \frac{x \cos x-\sin x}{x \sin ^{2} x}$$

Applications of Differentiation

Indeterminate Forms and L'Hopitals Rule

00:48

Calculus: Early Transcendental Functions

Find the indicated limits.
$$\lim _{x \rightarrow-2} \frac{x+2}{x^{2}-4}$$

Applications of Differentiation

Indeterminate Forms and L'Hopitals Rule

l'Hospital's Rule

145 Practice Problems

00:53

Calculus: Early Transcendental Functions

Find the indicated limits.
$$\lim _{x \rightarrow \infty} x \sin (1 / x)$$

Applications of Differentiation

Indeterminate Forms and L'Hopitals Rule

02:50

Calculus: Early Transcendentals

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable.
$$\lim _{\theta \rightarrow \pi / 2^{-}}(\tan \theta)^{\cos \theta}$$

Applications of the Derivative

L'Hôpital's Rule

01:53

Calculus: Early Transcendentals

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable.
$$\lim _{x \rightarrow \infty} \frac{x^{2}-\ln (2 / x)}{3 x^{2}+2 x}$$

Applications of the Derivative

L'Hôpital's Rule

Optimization Problems

249 Practice Problems

03:11

Calculus for Scientists and Engineers: Early Transcendental

A challenging pen problem Two triangular pens are built against a barn. Two hundred meters of fencing are to be used for the three sides and the diagonal dividing fence (see figure). What dimensions maximize the area of the pen? (FIGURE CAN'T COPY)

Applications of the Derivative

Optimization Problems

01:40

Calculus for Scientists and Engineers: Early Transcendental

Turning a corner with a pole
a. What is the length of the longest pole that can be carried horizontally around a corner at which a 3 -ft corridor and a 4 -ft corridor meet at right angles?
b. What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is $a$ feet wide and a corridor that is $b$ feet wide meet at right angles?
c. What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is $a=5 \mathrm{ft}$ wide and a corridor that is $b=5$ ft wide meet at an angle of $120^{\circ} ?$
d. What is the length of the longest pole that can be carried around a corner at which a corridor that is $a$ feet wide and a corridor that is $b$ feet wide meet at right angles, assuming there is an 8 -foot ceiling and that you may tilt the pole at any angle?

Applications of the Derivative

Optimization Problems

06:47

Calculus for Scientists and Engineers: Early Transcendental

Another pen problem A rancher is building a horse pen on the corner of her property using $1000 \mathrm{ft}$ of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure).
a. Determine the lengths of the sides that maximize the area of the pen.
b. Suppose there is already a fence along the side of the property opposite the side of length $y .$ Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing. (FIGURE CAN'T COPY)

Applications of the Derivative

Optimization Problems

Newton's Method

107 Practice Problems

01:49

Physics: A Conceptual World View

Explain how Newton's idea of light particles predicts that the speed of light in a transparent material will be faster than in a vacuum.

A Model for Light

02:33

Physics: A Conceptual World View

How does Newton's idea of light particles explain the law of refraction?

A Model for Light

03:21

Calculus for Scientists and Engineers: Early Transcendental

An eigenvalue problem A certain kind of differential equation (see Chapter 8 ) leads to the root-finding problem tan $\pi \lambda=\lambda$. where the roots $\lambda$ are called eigenvalues. Find the first three positive eigenvalues of this problem.

Applications of the Derivative

Newton's Method

Antiderivatives

240 Practice Problems

01:08

Calculus for Scientists and Engineers: Early Transcendental

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters.
$$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$

Applications of the Derivative

Antiderivatives

02:01

Calculus for Scientists and Engineers: Early Transcendental

Determine the following indefinite integrals. Check your work by differentiation.
$$\int \sqrt{x}\left(2 x^{6}-4 \sqrt[3]{x}\right) d x$$

Applications of the Derivative

Antiderivatives

01:42

Calculus for Scientists and Engineers: Early Transcendental

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. $F(x)=x^{3}-4 x+100$ and $G(x)=x^{3}-4 x-100$ are antiderivatives of the same function.
b. If $F^{\prime}(x)=f(x),$ then $f$ is an antiderivative of $F$
c. If $F^{\prime}(x)=f(x),$ then $\int f(x) d x=F(x)+C$
d. $f(x)=x^{3}+3$ and $g(x)=x^{3}-4$ are derivatives of the same function.
e. If $F^{\prime}(x)=G^{\prime}(x),$ then $F(x)=G(x)$

Applications of the Derivative

Antiderivatives

Rolle's Theorem

35 Practice Problems

01:32

Calculus: Early Transcendental Functions

For $$f(x)=\left\{\begin{array}{ll} 2 x & \text { if } x \leq 0 \\ 2 x-4 & \text { if } x>0 \end{array}\right.$$ show that $f$ is continuous on the interval $(0,2),$ differentiable on the interval (0,2) and has $f(0)=f(2) .$ Show that there does not exist a value of $c$ such that $f^{\prime}(c)=0 .$ Which hypothesis of Rolle's Theorem is not satisfied?

Differentiation

The Mean Value Theorem

03:24

Calculus: Early Transcendental Functions

Check the hypotheses of Rolle's Theorem and the Mean Value Theorem and find a value of $c$ that makes the appropriate conclusion true. Illustrate the conclusion with a graph.
$$f(x)=x^{3}+x^{2},[0,1]$$

Differentiation

The Mean Value Theorem

01:15

Calculus: Early Transcendentals

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle's Theorem.
$$f(x)=1-x^{2 / 3}:[-1,1]$$

Applications of the Derivative

Mean Value Theorem

Derivatives of Inverse Trigonmetric Functions

62 Practice Problems

01:41

Calculus for Scientists and Engineers: Early Transcendental

Identity proofs Prove the following identities and give the values of $x$ for which they are true.
$$\sin \left(2 \sin ^{-1} x\right)=2 x \sqrt{1-x^{2}}$$

Derivatives

Derivatives of Inverse Trigonometric Functions

00:32

Calculus for Scientists and Engineers: Early Transcendental

Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of $x,$ and find the derivative of the inverse function.
$$f(x)=\sqrt{x+2}, \text { for } x \geq-2$$

Derivatives

Derivatives of Inverse Trigonometric Functions

00:35

Calculus for Scientists and Engineers: Early Transcendental

Graphing $f$ and $f^{\prime}$.
a. Graph $f$ with a graphing utility.
b. Compute and graph $f^{\prime}$
c. Verify that the zeros of $f^{\prime}$ correspond to points at which $f$ has a horizontal tangent line.
$$f(x)=(x-1) \sin ^{-1} x \text { on }[-1,1]$$

Derivatives

Derivatives of Inverse Trigonometric Functions

Average Rates of Change and Secant Lines

31 Practice Problems

03:13

Calculus: Early Transcendental Functions

Use a CAS or graphing calculator.
Animate the secant lines in exercise $9,$ parts $(\mathrm{b}),(\mathrm{d})$ and $(\mathrm{f})$ converging to the tangent line in part (g).

Differentiation

Tangent Lines and Velocity

05:14

Calculus: Early Transcendentals

Explain the difference between the average rate of change and the instantancous rate of change of a function $f$

Derivatives

Derivatives as Rates of Change

03:42

Calculus: Early Transcendentals

Determine whether the following statements are true and give an explanation or counterexample.
a. For linear functions, the slope of any secant line always equals the slope of any tangent line.
b. The slope of the secant line passing through the points $P$ and $Q$ is less than the slope of the tangent line at $P$.
c. Consider the graph of the parabola $f(x)=x^{2} .$ For $a>0$ and $h>0,$ the secant line through $(a, f(a))$ and $(a+h, f(a+h))$ always has a greater slope than the tangent line at $(a, f(a))$

Derivatives

Introducing the Derivative

Instantaneous Rates of Change and Tangent Lines

108 Practice Problems

02:47

Applied Calculus

Figure 2.11 shows $N=f(t),$ the number of farms in the US $^{2}$ between 1930 and 2000 as a function of year, $t$
(a) Is $f^{\prime}(1950)$ positive or negative? What does this tell you about the number of farms?
(b) Which is more negative: $f^{\prime}(1960)$ or $f^{\prime}(1980) ?$ Explain.
(Check your book to see figure)

Rate of Change: The Derivative

Instantaneous Rate of Change

03:21

Precalculus

Find the limit of the difference quotient of the given function to obtain a function that represents the slope of a line drawn tangent to the curve at $x.$
$$f(x)=\frac{2}{x-1}$$

Bridges to Calculus: An Introduction to Limits

Applications of Limits: Instantaneous Rates of Change and the Area under a Curve

01:52

A Graphical Approach to Precalculus with Limits

Find the equation of the tangent line to the function $f$ at the given point. Then graph the function and the tangent line together.
$$f(x)=x-x^{2} \text { at }(-1,-2)$$

Limits, Derivatives, and Definite Integrals

Tangent Lines and Derivatives

Instantaneous Rates of Change and Tangent Lines: Estimating using Average Rate of Change

19 Practice Problems

03:30

Applied Calculus

A function $f$ has $f(5)=20, f^{\prime}(5)=2,$ and $f^{\prime \prime}(x)<0$ for $x \geq 5 .$ Which of the following are possible values for $f(7)$ and which are impossible?
(a) 26
(b) 24
(c) 22

Rate of Change: The Derivative

The Second Derivative

06:34

Applied Calculus

Values of $f(t)$ are given in the following table.
(a) Does this function appear to have a positive or negative first derivative? Second derivative? Explain.
(b) Estimate $f^{\prime}(2)$ and $f^{\prime}(8)$
$$\begin{array}{c|c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 & 10 \\\hline f(t) & 150 & 145 & 137 & 122 & 98 & 56 \\\hline\end{array}$$

Rate of Change: The Derivative

The Second Derivative

04:14

Applied Calculus

For the function $g(x)$ graphed in Figure $2.39,$ are the following nonzero quantities positive or negative?
(a) $g^{\prime}(0)$
(b) $g^{\prime \prime}(0)$
(GRAPH CAN'T COPY)

Rate of Change: The Derivative

The Second Derivative

Limit Definition of Derivative

77 Practice Problems

00:47

Calculus: Early Transcendental Functions

Graph $f(x)=|x|+|x-2|$ and identify all $x$ -values at which $f(x)$ is not differentiable.

Differentiation

The Derivative

03:52

Calculus: Early Transcendental Functions

Compute $f^{\prime}(a)$ using the limits (2.1) and (2.2).
$$f(x)=\frac{3}{x+1}, a=2$$

Differentiation

The Derivative

02:48

Calculus: Early Transcendentals

Vertical tangent lines If a function $f$ is continuous at a and $\lim _{x \rightarrow a}\left|f^{\prime}(x)\right|=\infty,$ then the curve $y=f(x)$ has a vertical tangent line at $a,$ and the equation of the tangent line is $x=a$. If $a$ is an endpoint of a domain, then the appropriate one-sided derivative (Exercises $71-72$ ) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. $x^{2}+y^{2}=9$
b. $x^{2}+y^{2}+2 x=0$

Derivatives

The Derivative as a Function

Derivative Rules

304 Practice Problems

01:34

Calculus: Early Transcendental Functions

Repeat example 5.1 by first substituting $x=t^{2}-1$ and $y=\sin t$ and then computing $g^{\prime}(t)$.

Functions of Several Variables and Partial Differentiation

The Chain Rule

01:27

Calculus: Early Transcendental Functions

Find the derivative of the expression for an unspecified differentiable function $f$.
$$\frac{\sqrt{x}}{f(x)}$$

Differentiation

The Product and Quotient Rules

04:48

Calculus: Early Transcendental Functions

For $f(x)=\sin x,$ find $f^{(05)}(x)$ and $f^{(150)}(x)$

Differentiation

Derivatives of Trigonometric Functions

Derivative Rules: Constant Multiple Rule

15 Practice Problems

03:10

Applied Calculus

In $2009,$ the population of Mexico was 111 million and growing $1.13 \%$ annually, while the population of the US was 307 million and growing $0.975 \%$ annually. $^{6}$ If we measure growth rates in people/year, which population was growing faster in 2009 ?

Shortcuts to Differentation

Exponential and Logarithmic Functions

02:15

Applied Calculus

With a yearly inflation rate of $5 \%,$ prices are given by
$$P=P_{0}(1.05)^{t}$$
where $P_{0}$ is the price in dollars when $t=0$ and $t$ is time in years. Suppose $P_{0}=1 .$ How fast (in cents/year) are prices rising when $t=10 ?$

Shortcuts to Differentation

Exponential and Logarithmic Functions

03:38

Applied Calculus

Find the equation of the tangent line to $f(x)=10 e^{-0.2 x}$ at $x=4.$

Shortcuts to Differentation

Exponential and Logarithmic Functions

Derivative Rules: Power Rule

32 Practice Problems

00:47

Calculus and Its Applications

Find $d y / d x .$ Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand.
$$y=\frac{x^{5}+x}{x^{2}}$$

Differentiation

Differentiation Techniques: The Power and Sum-Difference Rules

03:29

Calculus and Its Applications

Find an equation of the tangent line to the graph of $f(x)=\frac{1}{x^{2}}$
a) $\operatorname{at}(1,1)$
b) $\operatorname{at}\left(3, \frac{1}{9}\right)$
c) $\operatorname{at}\left(-2, \frac{1}{4}\right)$

Differentiation

Differentiation Techniques: The Power and Sum-Difference Rules

00:32

Calculus and Its Applications

Find $f^{\prime}(x)$.
$$f(x)=4 x-7$$

Differentiation

Differentiation Techniques: The Power and Sum-Difference Rules

Derivative Rules: Sum/Difference rule

41 Practice Problems

00:18

Calculus and Its Applications

For each of the following, graph $f$ and $f^{\prime}$ and then determine $f^{\prime}(1) .$ For Exercises use Deriv on the $T I-83$.
$$f(x)=\frac{4 x}{x^{2}+1}$$

Differentiation

Differentiation Techniques: The Power and Sum-Difference Rules

03:25

Applied Calculus

The yield, $Y$, of an apple orchard (measured in bushels of apples per acre) is a function of the amount $x$ of fertilizer in pounds used per acre. Suppose $$Y=f(x)=320+140 x-10 x^{2}$$
(a) What is the yield if 5 pounds of fertilizer is used per acre?
(b) Find $f^{\prime}(5) .$ Give units with your answer and interpret it in terms of apples and fertilizer.
(c) Given your answer to part (b), should more or less fertilizer be used? Explain.

Shortcuts to Differentation

Derivative Formulas for Powers and Polynomials

01:12

Calculus and Its Applications

For each function, find the interval(s) for which $f^{\prime}(x)$ is positive.
$$f(x)=x^{2}-4 x+1$$

Differentiation

Differentiation Techniques: The Power and Sum-Difference Rules

Derivative Rules: Product/Quotient Rule

62 Practice Problems

02:39

Calculus: Early Transcendental Functions

Use the quotient rule to show that the derivative of $[g(x)]^{-1}$ is $-g^{\prime}(x)[g(x)]^{-2} .$ Then use the product rule to compute the derivative of $f(x)[g(x)]^{-1}$.

Differentiation

The Product and Quotient Rules

01:52

Calculus: Early Transcendentals

Derivatives Find and simplify the derivative of the following functions.
$y=\frac{x-a}{\sqrt{x}-\sqrt{a}},$ where $a$ is a positive constant

Derivatives

The Product and Quotient Rules

05:21

Calculus: Early Transcendentals

Derivatives Find and simplify the derivative of the following functions.
$$h(x)=\frac{x+1}{x^{2} e^{x}}$$

Derivatives

The Product and Quotient Rules

Derivative Rules: Exponential/ Logarithm Rule

40 Practice Problems

01:10

Calculus: Early Transcendental Functions

Use logarithmic differentiation to find the derivative.
$$f(x)=x^{\sqrt{x}}$$

Differentiation

Derivatives of Exponential and Logarithmic Functions

05:40

Calculus: Early Transcendentals

Derivative of $u(x)^{\prime(x)}$ Use logarithmic differentiation to prove that
$$
\frac{d}{d x}\left(u(x)^{v(x)}\right)=u(x)^{v(x)}\left(\frac{d v}{d x} \ln u(x)+\frac{v(x)}{u(x)} \frac{d u}{d x}\right)
$$

Derivatives

Derivatives of Logarithmic and Exponential Functions

09:26