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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.

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

$g^\prime(1)$
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.
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 $\frac{5}{3}$ .

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

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

$3 · f^\prime(2)$

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

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