f(x)=8x^{5}(x^{4}-9)\nwhich of the following shows the correct application of the product rule?\na…

Explanation:

Step1: Recall product - rule

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

Step2: Find $u^\prime$

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

Step3: Find $v^\prime$

For $v=x^{4}-9$, $v^\prime=\frac{d}{dx}(x^{4}-9)=\frac{d}{dx}(x^{4})-\frac{d}{dx}(9)=4x^{3}-0 = 4x^{3}$.

Step4: Apply product - rule

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

Answer:

C. $(8x^{5})(4x^{3})+(x^{4}-9)(40x^{4})$