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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.7b
Chapter 3, Problem 3.10.7b

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
b. (f^-1)'(6)

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1
Identify the relationship between the function \( f \) and its inverse \( f^{-1} \). Recall that if \( y = f(x) \), then \( x = f^{-1}(y) \).
Use the formula for the derivative of an inverse function: \((f^{-1})'(b) = \frac{1}{f'(a)}\), where \( f(a) = b \).
From the problem, we need to find \((f^{-1})'(6)\). This means we need to find \( a \) such that \( f(a) = 6 \).
Look at the table provided to find the value of \( a \) for which \( f(a) = 6 \).
Once \( a \) is identified, find \( f'(a) \) from the table and use the formula \((f^{-1})'(6) = \frac{1}{f'(a)}\) to determine the derivative.

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Key Concepts

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

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x. Understanding how to find and interpret inverse functions is crucial for solving problems involving derivatives of these functions.
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Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f^-1)'(y) = 1 / f'(x), where y = f(x). This relationship shows that the rate of change of the inverse function at a point is the reciprocal of the rate of change of the original function at the corresponding point. This concept is essential for determining the derivative of f^-1 at a specific value.
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Using Tables for Derivatives

In calculus, tables can provide values of functions and their derivatives at specific points. When asked to find the derivative of an inverse function using a table, one must locate the corresponding values for f and f' to apply the inverse derivative formula. This method is particularly useful when explicit functions are not available.
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Related Practice
Textbook Question

31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

b. Determine the instantaneous velocity of the projectile at t=1 and t = 2 seconds.

s(t)= −16t²+100t

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For what values of x does g(x) = x−sin x have a slope of 1?

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109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limx→2(x2−3)5−1x−2{\(\displaystyle\]\lim\)_{x\(\to\)2}}\(\frac{\left(x^2-3\right)^5-1}{x-2}\)

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21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(s) = 4s³+3s; a= -3, -1

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Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

b. When does the stone reach its highest point?

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b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

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