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

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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2:16 minutes

Problem 3.R.13

Textbook Question

9–61. Evaluate and simplify y'.

y = e^2θ

Verified step by step guidance

Identify the given function: y = e^(2θ). We need to find the derivative y' with respect to θ.

Recognize that the function y = e^(2θ) is an exponential function where the exponent is a function of θ. This requires the use of the chain rule for differentiation.

Apply the chain rule: If y = e^(u) where u is a function of θ, then the derivative y' = e^(u) * (du/dθ). Here, u = 2θ.

Differentiate u = 2θ with respect to θ to find du/dθ. The derivative of 2θ with respect to θ is 2.

Substitute back into the chain rule formula: y' = e^(2θ) * 2. Simplify the expression to get the final form of the derivative.

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Key Concepts

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this case, we need to differentiate the function y = e^(2θ) with respect to θ to find y'.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is the base of the natural logarithm. These functions exhibit rapid growth or decay and are characterized by their constant rate of change. Understanding the properties of exponential functions is crucial for differentiating them correctly.

Chain Rule

The chain rule is a fundamental technique in calculus used 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 rule is essential when differentiating functions like y = e^(2θ), where the exponent is itself a function of θ.