find f(x). f(x)=(2x^6 + 1)^3 f(x)=

Explanation:

Step1: Identify the outer - inner functions

Let $u = 2x^{6}+1$, so $y = u^{3}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=3u^{2}$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=12x^{5}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2x^{6}+1$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. So $\frac{dy}{dx}=3(2x^{6}+1)^{2}\cdot12x^{5}$.

Step5: Simplify the result

$3(2x^{6}+1)^{2}\cdot12x^{5}=36x^{5}(2x^{6}+1)^{2}$.

Answer:

$36x^{5}(2x^{6}+1)^{2}$