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

Previous problemNext problem

2:45 minutes

Problem 3.8.14a

Textbook Question

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

Verified step by step guidance

Start by differentiating both sides of the equation with respect to x. The equation given is x = e^y.

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

Differentiate the right side: Use the chain rule. The derivative of e^y with respect to y is e^y, and then multiply by dy/dx because of the chain rule. So, the derivative is e^y * (dy/dx).

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

Solve for dy/dx by dividing both sides by e^y: dy/dx = 1/e^y. Substitute the point (2, ln 2) into the equation to find the specific value of dy/dx at that point.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule when necessary. This method is particularly useful for equations that define y implicitly in terms of x.

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 defined as a function of u, which in turn is a function of x, 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.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where a is a constant and x is the variable. In the context of the given problem, the equation x = e^y involves the natural exponential function, where e is the base of natural logarithms. Understanding the properties of exponential functions, including their derivatives, is crucial for applying implicit differentiation effectively.