find f(x) and simplify.\nf(x)=8x^{5}(x^{4}-9)\nwhich of the following shows the correct application of the…

find f(x) and simplify.\nf(x)=8x^{5}(x^{4}-9)\nwhich of the following shows the correct application of the product rule?\na. (8x^{5})(40x^{4})+(x^{4}-9)(4x^{3})\nb. (4x^{3})(40x^{4})\nc. (8x^{5})(4x^{3})+(x^{4}-9)(40x^{4})\nd. (8x^{5})(4x^{3})-(x^{4}-9)(40x^{4})

find f(x) and simplify.\nf(x)=8x^{5}(x^{4}-9)\nwhich of the following shows the correct application of the product rule?\na. (8x^{5})(40x^{4})+(x^{4}-9)(4x^{3})\nb. (4x^{3})(40x^{4})\nc. (8x^{5})(4x^{3})+(x^{4}-9)(40x^{4})\nd. (8x^{5})(4x^{3})-(x^{4}-9)(40x^{4})

Answer

Explanation:

Step1: Recall product - rule

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

Step2: Find (u^\prime(x))

Using the power - rule ((x^{n})^\prime = nx^{n - 1}), if (u(x)=8x^{5}), then (u^\prime(x)=8\times5x^{5 - 1}=40x^{4}).

Step3: Find (v^\prime(x))

If (v(x)=x^{4}-9), then (v^\prime(x)=4x^{4 - 1}=4x^{3}).

Step4: Apply product - rule

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

Answer:

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