Question

given that
$f(x)=x^{8}h(x)$
$h(-1)=3$
$h(-1)=6$

calculate $f(-1)$.
hint: use the product rule and the power rule.

Explanation:

Step1: Apply product - rule

The product rule states that if $f(x)=u(x)v(x)$, then $f'(x)=u'(x)v(x)+u(x)v'(x)$. Here, $u(x)=x^{8}$ and $v(x)=h(x)$. So, $f'(x)=(x^{8})'h(x)+x^{8}h'(x)$.

Step2: Differentiate $u(x)$ using power - rule

The power rule states that if $y = x^{n}$, then $y'=nx^{n - 1}$. For $u(x)=x^{8}$, $u'(x)=8x^{7}$. So, $f'(x)=8x^{7}h(x)+x^{8}h'(x)$.

Step3: Substitute $x=-1$

Substitute $x = - 1$ into $f'(x)$:

$$
LATEXBLOCK0
$$

Answer:

$-18$