Derivative of Inverse
To find the derivative of an inverse function, we can start with the formula for the derivative of a composition of functions
To find the derivative of an inverse function, we can start with the formula for the derivative of a composition of functions.
If we have a function f(x) and its inverse function g(x), then we have the following relationship between the two functions:
g(f(x)) = x
To find the derivative of g(x), we can differentiate both sides of this equation with respect to x:
d/dx [g(f(x))] = d/dx [x]
Using the chain rule on the left side, we get:
g'(f(x)) * f'(x) = 1
Now, solving for g'(f(x)), we get:
g'(f(x)) = 1/f'(x)
Therefore, the derivative of the inverse function g(x) is equal to 1 divided by the derivative of the original function f(x).
In mathematical notation, this can be written as:
(g^(-1))'(x) = 1/(f'(g(x)))
Note that in order to find the derivative of the inverse function, we need to know the derivative of the original function.
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