Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=tan x; (1,π/4)
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 3.7.28
27–76. Calculate the derivative of the following functions.
Verified step by step guidance1
Step 1: Recognize that the function y = (x^2 + 2x + 7)^8 is a composite function, which means we will use the chain rule to find its derivative.
Step 2: Identify the outer function and the inner function. Here, the outer function is u^8 where u = (x^2 + 2x + 7), and the inner function is u = x^2 + 2x + 7.
Step 3: Differentiate the outer function with respect to the inner function u. The derivative of u^8 with respect to u is 8u^7.
Step 4: Differentiate the inner function u = x^2 + 2x + 7 with respect to x. The derivative is 2x + 2.
Step 5: Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: dy/dx = 8(x^2 + 2x + 7)^7 * (2x + 2).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve of the function at any given point. Derivatives are used to find rates of change, optimize functions, and analyze the behavior of graphs.
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Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions raised to a power, as in the given problem.
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Power Rule
The power rule is a basic rule for finding the derivative of a function in the form of x raised to a power. It states that if f(x) = x^n, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions and is essential for solving problems involving powers, such as the one presented.
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Related Practice
Textbook Question
Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
Graph the following curves and determine the location of any vertical tangent lines.
a. x²+y² = 9
Textbook Question
If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.
Textbook Question
9–61. Evaluate and simplify y'.
y = (2x−3)x^3/2
Textbook Question
Find an equation of the line tangent to the curve y = sin x at x = 0.
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Textbook Question
Find d²/dx² (sin x + cos x).