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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:

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

$g^\prime(1)$
Hint (b):
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.
$\text{So, if }f\text{ has coordinate point (1, 2) and at }x=1, \text{ its derivative is }\frac{3}{5}.$
$\text{Then }g\text{ has coordinate point (2, 1) and at }x=2, \text{its derivative is }\frac{5}{3}.$

$g^\prime(2)$
Steps (c):
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).

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

$3 · f ^\prime(2)$

$5 · f^\prime(1) + 6 · g ^\prime(1)$
Hint (f):
Refer to the hint in part (b).

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