Book cover for Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein, David C. Lay, David I. Schneider

ISBN #9780134437774

14th Edition

3,033 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces advanced differentiation techniques including the Product Rule, Quotient Rule, and Chain Rule, which expand the ability to differentiate complex functions. In addition, the chapter covers implicit differentiation and related rates, both of which are essential for handling equations where variables are intertwined. Mastery of these rules is critical for solving a wide range of calculus problems effectively.

Learning Objectives

1

Explain and apply the Product and Quotient Rules to differentiate complex functions.

2

Understand and use the Chain Rule and General Power Rule to differentiate composite functions.

3

Demonstrate proficiency in implicit differentiation and related rates for problems where variables are interdependent.

4

Identify when and how to combine multiple differentiation techniques in a single problem.

Key Concepts

CONCEPT

DEFINITION

Product Rule

A differentiation rule used when differentiating the product of two functions. If f(x)=u(x)v(x), then f'(x)=u'(x)v(x)+u(x)v'(x).

Quotient Rule

A rule used when differentiating a function that is the quotient of two functions. For f(x)=u(x)/v(x), the derivative is given by f'(x)=[u'(x)v(x)-u(x)v'(x)]/v(x)^2.

Chain Rule

A powerful technique for differentiating composite functions. If a function y = f(g(x)), then y' = f'(g(x)) * g'(x).

General Power Rule

An extension of the power rule applied in conjunction with the chain rule. Used for functions of the form [g(x)]^n, the derivative is n[g(x)]^(n-1)*g'(x).

Implicit Differentiation

A method used to differentiate equations that define y implicitly as a function of x, rather than explicitly.

Related Rates

Problems that involve finding the rate at which one quantity changes in relation to another, typically using implicit differentiation.

Example Problems

Example 1

Differentiate the functions. $$y=(x+1)\left(x^{3}+5 x+2\right)$$

Example 2

Differentiate the functions. $$y=\left(-x^{3}+2\right)\left(\frac{x}{2}-1\right)$$

Example 3

Differentiate the functions. $$y=\left(2 x^{4}-x+1\right)\left(-x^{5}+1\right)$$

Example 4

Differentiate the functions. $$y=\left(x^{2}+x+1\right)^{3}(x-1)^{4}$$

Example 5

Differentiate the functions. $$y=x\left(x^{2}+1\right)^{4}$$
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Step-by-Step Explanations

QUESTION

Differentiate the function f(x) = (x^2 - 1)^4 (x^2 + 1)^5 using the product rule.

STEP-BY-STEP ANSWER:

Step 1: Identify u(x) = (x^2 - 1)^4 and v(x) = (x^2 + 1)^5.
Step 2: Differentiate u(x) using the chain rule: u'(x) = 4*(x^2 - 1)^3*(2x).
Step 3: Differentiate v(x) using the chain rule: v'(x) = 5*(x^2 + 1)^4*(2x).
Step 4: Apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x).
Step 5: Substitute the derivatives: f'(x) = [4*(x^2 - 1)^3*(2x)]*(x^2 + 1)^5 + (x^2 - 1)^4*[5*(x^2 + 1)^4*(2x)].
Final Answer: f'(x) = 8x*(x^2 - 1)^3*(x^2 + 1)^5 + 10x*(x^2 - 1)^4*(x^2 + 1)^4.

Product Rule

QUESTION

Differentiate g(x) = x^3 / (x^2 + 1)^4 using the quotient rule.

STEP-BY-STEP ANSWER:

Step 1: Let u(x) = x^3 and v(x) = (x^2 + 1)^4.
Step 2: Compute u'(x) = 3x^2.
Step 3: Differentiate v(x) using the chain rule: v'(x) = 4*(x^2 + 1)^3*(2x) = 8x*(x^2 + 1)^3.
Step 4: Apply the quotient rule: g'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]^2.
Step 5: Substitute values: g'(x) = [3x^2*(x^2 + 1)^4 - x^3*(8x*(x^2 + 1)^3)] / (x^2 + 1)^8.
Step 6: Simplify numerator: 3x^2*(x^2+1)^4 - 8x^4*(x^2+1)^3 = x^2*(x^2+1)^3[3*(x^2+1) - 8x^2].
Final Answer: g'(x) = x^2*(x^2+1)^3 [3*(x^2+1) - 8x^2] / (x^2+1)^8 = x^2[3*(x^2+1) - 8x^2] / (x^2+1)^5.

Quotient Rule

QUESTION

Differentiate h(x) = [(3x+2)^5] using the chain rule.

STEP-BY-STEP ANSWER:

Step 1: Identify the outer function f(u) = u^5 and the inner function u(x) = 3x + 2.
Step 2: Differentiate the outer function: f'(u) = 5u^4.
Step 3: Differentiate the inner function: u'(x) = 3.
Step 4: Apply the chain rule: h'(x) = f'(u(x)) * u'(x) = 5*(3x+2)^4 * 3.
Final Answer: h'(x) = 15*(3x+2)^4.

Chain Rule

QUESTION

Differentiate the equation x^2 + y^2 = 25 with respect to x.

STEP-BY-STEP ANSWER:

Step 1: Differentiate both sides of the equation: d/dx(x^2) + d/dx(y^2) = d/dx(25).
Step 2: Apply the power rule to x^2: 2x.
Step 3: Differentiate y^2: 2y*(dy/dx) (using the chain rule since y is a function of x).
Step 4: The derivative of 25 is 0.
Step 5: Write the equation: 2x + 2y*(dy/dx) = 0.
Step 6: Solve for dy/dx: 2y*(dy/dx) = -2x, so dy/dx = -x/y.
Final Answer: dy/dx = -x/y.

Implicit Differentiation

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Common Mistakes

  • Failing to apply the chain rule correctly when differentiating composite functions.
  • Omitting one component of the product or quotient rules, often leading to incorrect derivatives.
  • Sign errors while applying the quotient rule, particularly in the numerator.
  • Neglecting to differentiate all terms in an implicit function, leading to incomplete solutions.