# How do you find the derivative of the inverse of f(x)=7x+6 using the definition of the derivative of an inverse function?

Jan 6, 2017

Please see the explanation section below.

#### Explanation:

I'm not sure what you mean by the "definition of the derivative of an inverse function".

The definition of the derivative of any function is the same. We can write the definition using inverse function notation as

$\frac{d}{\mathrm{dx}} \left({f}^{-} 1 \left(x\right)\right) = {\lim}_{h \rightarrow 0} \frac{{f}^{-} 1 \left(x + h\right) - {f}^{-} 1 \left(x\right)}{h}$,

but that's just notation.

I'm guessing that you want to use a Theorem about derivatives of inverse functions.

$\frac{d}{\mathrm{dx}} \left({f}^{-} 1 \left(x\right)\right) = \frac{1}{f ' \left({f}^{-} 1 \left(x\right)\right)}$

In this case

$f \left(x\right) = 7 x + 6$, so $f ' \left(x\right) = 7$.

This means that $f ' \left({f}^{-} 1 \left(x\right)\right) = 7$ for all $x$.

Therefore

$\frac{d}{\mathrm{dx}} \left({f}^{-} 1 \left(x\right)\right) = \frac{1}{7}$.