Question

given that (f(x)=x^{7}h(x)), (h(-1) = 4), and (h(-1)=7), calculate (f(-1)).

(f(-1)=)

question help: video

Explanation:

Step1: Apply product - rule

The product - rule states that if $f(x)=u(x)v(x)$, then $f^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)$. Here, $u(x)=x^{7}$ and $v(x)=h(x)$. So, $u^{\prime}(x) = 7x^{6}$ and $v^{\prime}(x)=h^{\prime}(x)$. Then $f^{\prime}(x)=7x^{6}h(x)+x^{7}h^{\prime}(x)$.

Step2: Substitute $x = - 1$

Substitute $x=-1$, $h(-1) = 4$, and $h^{\prime}(-1)=7$ into $f^{\prime}(x)$.
When $x=-1$, we have $f^{\prime}(-1)=7(-1)^{6}h(-1)+(-1)^{7}h^{\prime}(-1)$.
Since $(-1)^{6}=1$ and $(-1)^{7}=-1$, then $f^{\prime}(-1)=7\times1\times4+(-1)\times7$.

Step3: Calculate the value

$f^{\prime}(-1)=28 - 7=21$.

Answer:

$21$