Textbook Question
9–61. Evaluate and simplify y'.
y = e^sin x+2x+1
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.R.28
9–61. Evaluate and simplify y'.
y = e^tan x (tan x−1)
Verified step by step guidance1
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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3mKey Concepts
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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.
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05:53
Finding Differentials
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.
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05:18The Product Rule
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.
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05:02Intro to the Chain Rule
Related Practice
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Textbook Question
{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.
b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.