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Crazy boy [7]
2 years ago
6

You can find the derivative of the inverse function by differentiating the inverse explicitly, or by using the formula:

Mathematics
1 answer:
Dafna11 [192]2 years ago
4 0

If a function f(x) has an inverse f ⁻¹(x), then by definition

f\left(f^{-1}(x)\right) = x

Differentiating both sides with respect to x yields

f'\left(f^{-1}(x)\right) \times \left(f^{-1}\right)'(x) = 1 \implies \left(f^{-1}\right)'(x) = \dfrac1{f'\left(f^{-1}(x)\right)}

Now, if f(a) = b and b = f ⁻¹(a), then

\left(f^{-1}\right)'(a) = \dfrac1{f'\left(f^{-1}(a)\right)} = \dfrac1{f'(b)}

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