### Home > APCALC > Chapter 7 > Lesson 7.3.4 > Problem7-148

7-148.

Let $f$ and $g$ be continuous and differentiable such that $f(g(x)) = x$.

 $x$ $f(x)$ $f ^\prime(x)$ $g(x)$ $–1$ $0$ $1$ $2$ $2$ $–1$ $1$ $0$ $1$ $2$ $7$ $3$ $0$ $2$ $1$ $–1$

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.

$\text{So, if }f\text{ has coordinate point (1, 2) and at }x=1, \text{ its derivative is }\frac{3}{5}.$

$\text{Then }g\text{ has coordinate point (2, 1) and at }x=2, \text{its derivative is }\frac{5}{3}.$

1. $g^\prime(2)$

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

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

1. $3 · f ^\prime(2)$

1. $5 · f^\prime(1) + 6 · g ^\prime(1)$

Refer to the hint in part (b).