Question

differentiate the function after first rewriting the function in a different form. (do not use the product or quotient rules.)
$f(x)=x^{3}(x + 8)$
$f(x)=$

Explanation:

Step1: Expand the function

$f(x)=x^{3}(x + 8)=x^{4}+8x^{3}$

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For $y=x^{4}$, $y^\prime = 4x^{3}$; for $y = 8x^{3}$, $y^\prime=8\times3x^{2}=24x^{2}$.

Step3: Find the derivative of the sum

$f^\prime(x)=(x^{4}+8x^{3})^\prime=(x^{4})^\prime+(8x^{3})^\prime=4x^{3}+24x^{2}$

Answer:

$4x^{3}+24x^{2}$