Question

find f(x). f(x) = 2x^{4}ln x f(x)=square

Explanation:

Step1: Recall product - rule

The product - rule states that if $y = u\cdot v$, then $y^\prime=u^\prime v + uv^\prime$. Here, $u = 2x^{4}$ and $v=\ln x$.

Step2: Differentiate $u = 2x^{4}$

Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $u^\prime=\frac{d}{dx}(2x^{4})=2\times4x^{4 - 1}=8x^{3}$.

Step3: Differentiate $v=\ln x$

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

Step4: Apply product - rule

$f^\prime(x)=u^\prime v+uv^\prime=(8x^{3})\ln x+(2x^{4})\times\frac{1}{x}$.

Step5: Simplify the expression

$f^\prime(x)=8x^{3}\ln x + 2x^{3}=2x^{3}(4\ln x + 1)$.

Answer:

$2x^{3}(4\ln x + 1)$