Question

find f(x).
f(x) = (x^2 - 7)(x^2 + 6)
f(x) =
(type an exact answer.)

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u=x^{2}-7$ and $v = x^{2}+6$.

Step2: Find $u'$ and $v'$

Differentiate $u=x^{2}-7$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $u'=\frac{d}{dx}(x^{2}-7)=2x$. Differentiate $v=x^{2}+6$ with respect to $x$, we get $v'=\frac{d}{dx}(x^{2}+6)=2x$.

Step3: Calculate $f'(x)$

Substitute $u$, $v$, $u'$, and $v'$ into the product - rule formula:
$f'(x)=u'v+uv'=(2x)(x^{2}+6)+(x^{2}-7)(2x)$.
Expand the expression:
\[

$$\begin{align*} f'(x)&=2x\cdot x^{2}+2x\cdot6 + 2x\cdot x^{2}-2x\cdot7\\ &=2x^{3}+12x + 2x^{3}-14x\\ &=(2x^{3}+2x^{3})+(12x - 14x)\\ &=4x^{3}-2x \end{align*}$$

\]

Answer:

$4x^{3}-2x$