Question

find f(x).
f(x)=3x^{5}ln x
f(x)=□

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Here, $u = 3x^{5}$ and $v=\ln x$.

Step2: Find $u^\prime$

Differentiate $u = 3x^{5}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u^\prime=\frac{d}{dx}(3x^{5})=15x^{4}$.

Step3: Find $v^\prime$

Differentiate $v=\ln x$ with respect to $x$. We know that $\frac{d}{dx}(\ln x)=\frac{1}{x}$.

Step4: Calculate $f^\prime(x)$

Substitute $u$, $v$, $u^\prime$, and $v^\prime$ into the product - rule formula: $f^\prime(x)=u^\prime v+uv^\prime=15x^{4}\ln x + 3x^{5}\cdot\frac{1}{x}$.
Simplify the second term: $3x^{5}\cdot\frac{1}{x}=3x^{4}$.
So, $f^\prime(x)=15x^{4}\ln x + 3x^{4}=3x^{4}(5\ln x + 1)$.

Answer:

$3x^{4}(5\ln x + 1)$