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)=4e^10x; (4,0)
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Inverse Trigonometric Functions
2:00 minutesProblem 3.10.67
Textbook Question
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = 3x-4
Verified step by step guidance1
First, understand that if f(x) is a function, then its inverse, denoted as f^(-1)(x), is a function such that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.
To find the derivative of the inverse function, we use the formula: (f^(-1))'(y) = 1 / f'(x), where y = f(x).
Given the function f(x) = 3x - 4, we first need to find its derivative, f'(x). Differentiate f(x) with respect to x to get f'(x) = 3.
Now, apply the formula for the derivative of the inverse function: (f^(-1))'(y) = 1 / f'(x). Since f'(x) = 3, we have (f^(-1))'(y) = 1 / 3.
Thus, the derivative of the inverse function f^(-1)(x) is a constant value, 1/3, for all x in the domain of the inverse function.
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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 essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and work with inverse functions is crucial for solving problems involving derivatives of these functions.
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Derivative of a Function
The derivative of a function measures how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. This concept is fundamental in calculus, as it provides the slope of the tangent line to the function at any given point.
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Derivative of Inverse Functions Theorem
The Derivative of Inverse Functions Theorem states that if f is a differentiable function and f'(x) is non-zero, then the derivative of its inverse function f⁻¹ at a point y is given by f⁻¹'(y) = 1 / f'(f⁻¹(y)). This theorem is essential for finding the derivative of an inverse function, as it connects the derivatives of the original and inverse functions.
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