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# System prepended metadata

title: Inverse Derivative formula

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# Inverse Derivative formula

:::info
**Prove.** Given inverse functions $f(x)$ and $g(x)$, the derivative $g^\prime(x)$ is given by

$$
g^\prime(x) = \frac{1}{f^\prime(g(x))}
$$
:::

**Proof.** Let $f(x)$ and $g(x)$ be inverse functions so that,
$$f(g(x)) = x.$$

Notice that the derivative of both sides yields,

$$
\begin{align*}
\frac{\text{d}}{\text{d}x}\bigg[f(g(x))\bigg] &= \frac{\text{d}}{\text{d}x}\big[x\big] \\
f^\prime(g(x)) \cdot g^\prime(x) &= 1 \\
g^\prime(x) &= \frac{1}{f^\prime(g(x))}
\end{align*}
$$