Question

find f(x). f(x)=5x^{3}(x^{2}-5) f(x)=

Explanation:

Step1: Use the product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Let $u = 5x^{3}$ and $v=x^{2}-5$. First, find $u'$ and $v'$.
$u'=\frac{d}{dx}(5x^{3})=15x^{2}$ (using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$), and $v'=\frac{d}{dx}(x^{2}-5)=2x$.

Step2: Apply the product - rule formula

$f'(x)=u'v + uv'$. Substitute $u = 5x^{3}$, $u' = 15x^{2}$, $v=x^{2}-5$, and $v' = 2x$ into the formula.
$f'(x)=15x^{2}(x^{2}-5)+5x^{3}(2x)$.

Step3: Expand the expressions

$15x^{2}(x^{2}-5)=15x^{4}-75x^{2}$ and $5x^{3}(2x)=10x^{4}$.

Step4: Combine like terms

$f'(x)=(15x^{4}-75x^{2})+10x^{4}=25x^{4}-75x^{2}$.

Answer:

$25x^{4}-75x^{2}$