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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2:36 minutes

Problem 3.R.26

Textbook Question

9–61. Evaluate and simplify y'.

y = e^sin x+2x+1

Verified step by step guidance

First, identify the function y given in the problem. Here, y = e^(sin(x)) + 2x + 1.

Next, we need to find the derivative of y with respect to x, denoted as y'. This involves differentiating each term in the function y separately.

For the term e^(sin(x)), use the chain rule. The derivative of e^(u) with respect to u is e^(u), and the derivative of sin(x) with respect to x is cos(x). Therefore, the derivative of e^(sin(x)) is e^(sin(x)) * cos(x).

For the term 2x, the derivative is straightforward. The derivative of 2x with respect to x is simply 2.

The derivative of the constant term 1 is 0, as constants have no rate of change. Combine all these derivatives to express y' as y' = e^(sin(x)) * cos(x) + 2.

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Key Concepts

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

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In this context, we need to apply differentiation rules to the function y = e^(sin x) + 2x + 1 to find y'.

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 is particularly relevant for the term e^(sin x) in the given function.

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where a is a constant. In this case, e^(sin x) is an exponential function where the exponent is a function itself (sin x). Understanding how to differentiate exponential functions is crucial for evaluating y' accurately.

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9–61. Evaluate and simplify y'.y = e^sin x+2x+1

Key Concepts

Differentiation

Chain Rule

Exponential Functions

Watch next

9–61. Evaluate and simplify y'.

y = e^sin x+2x+1