Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

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2:55 minutes

Problem 3.8.63a

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
a. Use implicit differentiation to find dy/dx.

Verified step by step guidance

Start by differentiating both sides of the equation y(x²+4)=8 with respect to x. This involves using implicit differentiation, where you treat y as a function of x.

Apply the product rule to differentiate the left side of the equation. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx), where u and v are functions of x. Here, u = y and v = (x²+4).

Differentiate u = y with respect to x, which gives dy/dx. Differentiate v = (x²+4) with respect to x, which gives 2x.

Substitute the derivatives back into the product rule: y(2x) + (x²+4)(dy/dx) = 0. This equation comes from differentiating the left side of the original equation.

Solve for dy/dx by isolating it on one side of the equation. Rearrange the terms to get (x²+4)(dy/dx) = -y(2x), and then divide both sides by (x²+4) to find dy/dx = -y(2x)/(x²+4).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated on one side. Instead of solving for y explicitly, we differentiate both sides of the equation with respect to x, applying the chain rule when necessary. This method is particularly useful for curves defined by equations that cannot be easily rearranged.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation, where y is often a function of x indirectly.

Witch of Agnesi

The Witch of Agnesi is a specific type of curve defined by the equation y(x² + 4) = 8, which can be rearranged to express y in terms of x. This curve is notable in mathematics for its unique shape and properties, resembling a bell. Understanding its equation and characteristics is crucial for applying differentiation techniques effectively in the context of the problem.

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Related Practice

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

x+y³−y=1; x=1

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

x+y³−y=1; x=1

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

4x³ =y²(4−x); x=2 (cissoid of Diocles)

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

b. Graph the tangent lines on the given graph.

4x³ =y²(4−x); x=2 (cissoid of Diocles)

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

Textbook Question

Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

d. Verify that the results of parts (a) and (c) are consistent.

Textbook Question

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (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. <IMAGE>

Exercise 46

Textbook Question

Exercise 48

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