Question

find f(x) and simplify.
f(x)=8x^5(x^4 - 9)
which of the following shows the correct application of the product rule?
a. (8x^5)(40x^4)+(x^4 - 9)(4x^3)
b. (4x^3)(40x^4)
c. (8x^5)(4x^3)+(x^4 - 9)(40x^4)
d. (8x^5)(4x^3)-(x^4 - 9)(40x^4)
f(x)=□

Explanation:

Step1: Recall product - rule

The product rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u = 8x^{5}$ and $v=x^{4}-9$.

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

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $u = 8x^{5}$, $u'=8\times5x^{4}=40x^{4}$, and for $v=x^{4}-9$, $v' = 4x^{3}$.

Step3: Apply product - rule

$f'(x)=u'v+uv'=(40x^{4})(x^{4}-9)+(8x^{5})(4x^{3})$.

Step4: Expand and simplify

$$
LATEXBLOCK0
$$

Answer:

$72x^{8}-360x^{4}$