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 $f'(x)$:
\[

$$\begin{align*} f'(-1)&=8(-1)^{7}h(-1)+(-1)^{8}h'(-1)\\ &=8\times(-1)\times3 + 1\times6\\ &=-24 + 6\\ &=-18 \end{align*}$$

\]

Answer:

$-18$