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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1:48 minutes

Problem 3.8.19a

Textbook Question

13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
cos y = x; (0, π/2)

Verified step by step guidance

Start by differentiating both sides of the equation with respect to x. The equation is cos(y) = x.

Differentiate the left side: The derivative of cos(y) with respect to x is -sin(y) * (dy/dx) because of the chain rule.

Differentiate the right side: The derivative of x with respect to x is 1.

Set the derivatives equal to each other: -sin(y) * (dy/dx) = 1.

Solve for dy/dx by isolating it: dy/dx = -1/sin(y).

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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 explicitly separated. Instead of solving for y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method is particularly useful for equations that are difficult or impossible to rearrange.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, the chain rule is applied to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating a function of y, we multiply by dy/dx to reflect the dependence of y on x.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are essential in calculus for modeling periodic phenomena and solving various problems. In the context of the given equation, cos(y) = x, understanding the properties and derivatives of these functions is crucial for finding dy/dx. The derivative of cos(y) involves the chain rule and is equal to -sin(y) * dy/dx.