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

Product and Quotient Rules

Calculus

3. Techniques of Differentiation

Product and Quotient Rules

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3:58 minutes

Problem 3.R.28

Textbook Question

9–61. Evaluate and simplify y'.

y = e^tan x (tan x−1)

Verified step by step guidance

First, identify the function y given in the problem. Here, y = e^(tan(x)) * (tan(x) - 1).

To find y', the derivative of y with respect to x, apply the product rule. The product rule states that if you have a function u(x) * v(x), then the derivative is u'(x) * v(x) + u(x) * v'(x).

Let u(x) = e^(tan(x)) and v(x) = tan(x) - 1. Find the derivative of u(x), which involves the chain rule. The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).

Calculate u'(x): The derivative of e^(tan(x)) is e^(tan(x)) * sec^2(x), because the derivative of tan(x) is sec^2(x).

Calculate v'(x): The derivative of tan(x) - 1 is sec^2(x), since the derivative of tan(x) is sec^2(x) and the derivative of a constant is 0.

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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 given function y = e^(tan x) (tan x - 1) to find y'. This involves using the product rule and the chain rule, as the function is a product of two functions.

Product Rule

The product rule is a formula used to differentiate products of two functions. It states that if you have two functions u(x) and v(x), the derivative of their product is given by u'v + uv'. In the case of y = e^(tan x) (tan x - 1), we will apply the product rule to differentiate the two components of the function.

Chain Rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. In this problem, we will use the chain rule to differentiate e^(tan x) since tan x is itself a function of x.