Derivatives Of Inverse Functions Homework Answers
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How to Find the Derivatives of Inverse Functions: A Step-by-Step Guide
Inverse functions are functions that undo each other, such as f(x) = e^x and g(x) = ln(x). Finding the derivatives of inverse functions can be tricky, but there is a formula that makes it easier. In this article, we will explain how to use the formula and apply it to some examples.
The Formula for Derivatives of Inverse Functions
The formula for finding the derivative of an inverse function is:
(f^-1)'(x) = 1 / f'(f^-1(x))
This means that the derivative of the inverse function f^-1(x) at a point x is equal to the reciprocal of the derivative of the original function f(x) at the point f^-1(x).
To use this formula, we need to know two things: the inverse function f^-1(x) and the derivative of the original function f'(x).
Examples of Derivatives of Inverse Functions
Let's see how to use the formula on some examples.
Example 1: f(x) = x + 2 / x
The inverse of f(x) is g(x) = 2x - 1 / x. To find g'(x), we use the formula:
g'(x) = 1 / f'(g(x))
We first need to find f'(x), which is:
f'(x) = (x - 2) / x^2
Then we plug in g(x) into f'(x), which gives us:
f'(g(x)) = (g(x) - 2) / g(x)^2
f'(g(x)) = ((2x - 1) / x - 2) / ((2x - 1) / x)^2
f'(g(x)) = (x^2 - 4x + 1) / (4x^2 - 4x + 1)
Finally, we take the reciprocal of f'(g(x)) to get g'(x):
g'(x) = (4x^2 - 4x + 1) / (x^2 - 4x + 1)
Example 2: f(x) = sin(x)
The inverse of f(x) is g(x) = arcsin(x). To find g'(x), we use the formula:
g'(x) = 1 / f'(g(x))
We first need to find f'(x), which is:
f'(x) = cos(x)
Then we plug in g(x) into f'(x), which gives us:
f'(g(x)) = cos(arcsin(x))
We can use a trigonometric identity to simplify this expression:
cos(arcsin(x)) = sqrt(1 - x^2)
Finally, we take the reciprocal of f'(g(x)) to get g'(x):
g'(x) = 1 / sqrt(1 - x^2)
Example 3: f(x) = x^3
The inverse of f(x) is g(x) = x^(1/3). To find g'(x), we use the formula:
g'(x) = 1 / f'(g(x))
We first need to find f'(x), which is:
f'(x) = 3x^2
Then we plug in g(x) into f'(x), which gives us:
f'(g(x)) = 3(g(x))^2
f'(g(x)) = 3(x^(1/3))^2
f'(g(x)) = 3x^(2/3)
Finally, we take the reciprocal of f'(g(x)) to get g'(x):
g'(x) = 1 / 3x^(2/3)
Example 4: f(x) = tan(x)
The inverse of f(x) is g(x) = arctan(x). To find g'(x), we use the formula:
g'(x) = 1 / f'(g(x))
We first need to find f'(x), which is:
f'(x) = sec^2(x)
Then we plug in g(x) into f'(x), which gives us:
f'(g(x)) = sec^2(arctan(x))
We can use a trigonometric identity to simplify this expression:
sec^2(arctan(x)) = 1 + tan^2(arctan(x))
sec^2(arctan(x)) = 1 + x^2
Finally, we take the reciprocal of f'(g(x)) to get g'(x):
g'(x) = 1 / (1 + x^2) a474f39169