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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.10.7b
Capitolo 3, Problema 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)

Guida verificata passo dopo passo
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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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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Inverse Cosine

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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Derivatives of Inverse Sine & Inverse Cosine

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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Derivatives
Pratica correlata
Domanda del libro di testo

A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>

b. Which runner has the greater average angular velocity?

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Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.

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Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

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45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

x⁴-x²y+y⁴=1; (−1, 1)

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21–30. Derivatives

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

f(t) = 3t⁴; a= -2, 2

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45–50. Tangent lines Carry out the following steps. <IMAGE>

b. Determine an equation of the line tangent to the curve at the given point.

(x²+y²)²=25/4 xy²; (1, 2)

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