### Home > CALC > Chapter 7 > Lesson 7.3.3 > Problem7-137

7-137.

Given: $f(x)$ and $g(x)$ continuous and differentiable such that $f(g(x)) = x$.

$x$

$f(x)$

$f^\prime(x)$

$g(x)$

$-1$

$2$

$1$

$0$

$0$

$-1$

$2$

$2$

$1$

$1$

$7$

$1$

$2$

$0$

$3$

$-1$

Evaluate.

1. $f^\prime(g(0))$

1. $g^\prime(1)$

How to find derivative of inverse functions:
A function,$f$, and its inverse, $g$, will have reciprocal derivatives at corresponding $(x, y)→(y, x)$ values.
So, if $f$ has coordinate point $(1, 2)$ and at $x=1$, its derivative is $\frac{3}{5}$.
Then $g$ has coordinate point $(2, 1)$ and at $x=2$, its derivative is $\frac{5}{3}$.

1. $g^\prime(2)$

Since $g(2) = −1$, the inverse, $f(−1) = 2$.
Now $f^\prime(−1) = 1$, so $g^\prime(2) =$ __
Refer to the hint in part (b)

1. $f^\prime(g(2))$

1. $3 · f^\prime(2)$

1. $5 · f^\prime(1) + 6 · g ′(1)$

Refer to the hint in part (b).