Question

1. if $h(x)=f(x)^{2}-g(x)^{2}$, $f(x)=-g(x)$, and $g(x)=f(x)$, then $h(x)=$

a. 0
b. $-4f(x)g(x)$
c. $(-g(x))^{2}-(f(x))^{2}$
d. $-2(f(x)-g(x))$

Explanation:

Step1: Apply chain rule to $h(x)$

$h'(x) = 2f(x)f'(x) - 2g(x)g'(x)$

Step2: Substitute given derivatives

$h'(x) = 2f(x)(-g(x)) - 2g(x)(f(x))$

Step3: Simplify the expression

$h'(x) = -2f(x)g(x) - 2f(x)g(x) = -4f(x)g(x)$

Answer:

B. $-4f(x)g(x)$