find f(x). f(x)=3x^4 ln x f(x)=□

Explanation:

Step1: Apply product - rule

The product - rule states that if $y = u\cdot v$, then $y'=u'v + uv'$. Here, $u = 3x^{4}$ and $v=\ln x$.

Step2: Find $u'$

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, for $u = 3x^{4}$, we have $u'=\frac{d}{dx}(3x^{4})=3\times4x^{3}=12x^{3}$.

Step3: Find $v'$

The derivative of $\ln x$ with respect to $x$ is $\frac{1}{x}$, so $v'=\frac{1}{x}$.

Step4: Calculate $f'(x)$

By the product - rule $f'(x)=u'v + uv'$. Substitute $u = 3x^{4}$, $u' = 12x^{3}$, $v=\ln x$, and $v'=\frac{1}{x}$ into the formula: $f'(x)=12x^{3}\ln x+3x^{4}\times\frac{1}{x}=12x^{3}\ln x + 3x^{3}$.

Answer:

$12x^{3}\ln x+3x^{3}$