Textbook Question
Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec−1(ex); (ln 2,π/3)
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
3:06 minutesProblem 3.10.7c
Textbook Question
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>
c. (f^-1)'(1)
Verified step by step guidance1
Identify the relationship between a function and its inverse. If \( y = f(x) \), then \( x = f^{-1}(y) \). The derivative of the inverse function \( (f^{-1})'(y) \) can be found using the formula \( (f^{-1})'(y) = \frac{1}{f'(x)} \) where \( x = f^{-1}(y) \).
From the problem, we need to find \( (f^{-1})'(1) \). This means we need to find the value of \( x \) such that \( f(x) = 1 \).
Look at the table provided in the problem to find the value of \( x \) for which \( f(x) = 1 \). This will give us the point \( (x, 1) \) on the graph of \( f \).
Once the correct \( x \) is identified, use the table to find \( f'(x) \), the derivative of \( f \) at this \( x \).
Finally, apply the formula \( (f^{-1})'(1) = \frac{1}{f'(x)} \) using the value of \( f'(x) \) obtained from the table to find the derivative of the inverse function at the given point.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
Inverse functions are functions that 'reverse' 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 work with inverse functions is crucial for determining their derivatives.
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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 derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept is essential for solving problems involving derivatives of inverse functions.
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Using Tables for Derivatives
When working with derivatives from tables, it is important to locate the necessary values for the function and its derivative. The table typically provides values of f(x) and f'(x) at specific points, which can be used to find the derivative of the inverse function. Understanding how to interpret and extract information from these tables is key to solving derivative problems.
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