Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section provides a comprehensive foundation in differentiation rules including constant functions, power functions, and exponential functions. It demonstrates how to derive these rules from first principles and apply them to compute derivatives quickly. The rules such as the Power Rule, Product Rule, and Sum/Difference Rules, not only simplify computing derivatives but also have practical applications in finding tangent and normal lines and modeling real-world phenomena like motion.

Learning Objectives

1

Differentiate constant functions, power functions, polynomials, and exponential functions using differentiation rules.

2

Apply the Power Rule, Constant Multiple Rule, and Sum/Difference Rules to compute derivatives of various functions.

3

Utilize the definition of the derivative to verify differentiation rules, including for negative and fractional exponents.

4

Determine equations of tangent and normal lines using derivatives and interpret their geometric meaning.

5

Understand and derive the Product Rule, and appreciate the significance of the number e in natural exponential functions.

Key Concepts

CONCEPT

DEFINITION

Derivative

The derivative of a function at a point is the limit of the difference quotient and represents the instantaneous rate of change or slope of the tangent line at that point.

Constant Function

A function of the form f(x) = c, where c is a constant. Its derivative is 0 because the graph is a horizontal line.

Power Rule

A rule stating that if f(x) = x^n for any real number n, then f'(x) = n*x^(n-1).

Exponential Function

A function of the form f(x) = b^x. In particular, when b = e (Euler’s number), we have the natural exponential function with the unique property that its derivative is itself.

Product Rule

A rule for differentiating the product of two differentiable functions: if u(x) and v(x) are differentiable, then (uv)' = u'v + uv'.

Constant Multiple Rule

A rule stating that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.

Sum/Difference Rule

A rule stating that the derivative of the sum (or difference) of functions is the sum (or difference) of their derivatives.

Number e

The unique real number such that the derivative of e^x is e^x, defined by the limit lim (h→0)(e^h - 1)/h = 1.

Example Problems

Example 1

(a) How is the number $ e $ defined? (b) Use a calculator to estimate the values of the limits $ \displaystyle \lim_{h\to 0}\frac {2.7^h - 1}{h} $ and $ \displaystyle \lim_{h\to 0}\frac {2.8^h - 1}{h} $ correct to two decimal places. What can you conclude about the value of $ e $?

Example 2

(a) Sketch, by hand, the graph of the function $ f(x) = e^x, $ paying particular attention to the graph crosses the y-axes. What fact allows you to do this? (b) What types of functions are $ f(x) = e^x $ and $ g(x) = x^e? $ Compare the differentiation formulas for $ f $ and $ g. $ (c) Which of the two functions in part (b) grows more rapidly when $ x $ is large?

Example 3

Differentiate the function. $ f(x) = 2^{40} $

Example 4

Differentiate the function. $ f(x) = e^5 $

Example 5

Differentiate the function. $ f(x) = 5.2x + 2.3 $
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Step-by-Step Explanations

QUESTION

Show that the derivative of f(x) = c is 0 using the definition of the derivative.

STEP-BY-STEP ANSWER:

Step 1: Write the definition: f'(x) = lim (h→0) [f(x+h) - f(x)]/h.
Step 2: Substitute f(x) = c into the expression: lim (h→0) [c - c]/h.
Step 3: Simplify the numerator: c - c = 0.
Step 4: The limit becomes lim (h→0) 0/h = 0.
Final Answer: f'(x) = 0.

Derivative of a Constant Function

QUESTION

Use the Power Rule to differentiate f(x) = x^4.

STEP-BY-STEP ANSWER:

Step 1: Identify the exponent n = 4.
Step 2: Apply the Power Rule: f'(x) = n*x^(n-1).
Step 3: Substitute values to get: f'(x) = 4*x^(4-1) = 4x^3.
Final Answer: f'(x) = 4x^3.

Power Rule for f(x) = x^4

QUESTION

Show that the derivative of the natural exponential function f(x) = e^x is e^x.

STEP-BY-STEP ANSWER:

Step 1: Write the definition: f'(x) = lim (h→0) [e^(x+h) - e^x]/h.
Step 2: Factor out e^x to obtain: e^x * lim (h→0) [e^h - 1]/h.
Step 3: Recognize that lim (h→0) [e^h - 1]/h = 1 (by definition of e).
Step 4: Conclude that f'(x) = e^x * 1 = e^x.
Final Answer: The derivative of e^x is e^x.

Derivative of f(x) = e^x

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Common Mistakes

  • Mistake: Assuming the derivative of a constant is not zero.
  • Misconception: Believing that the derivative of a product is the product of the derivatives, rather than applying the Product Rule.
  • Error: Incorrectly handling negative and fractional exponents when applying the Power Rule.
  • Mistake: Forgetting to multiply by the constant in the Constant Multiple Rule.
  • Misconception: Overlooking the importance of verifying the differentiability at all critical points, such as where the function may not be smooth.