find the derivatives of the following functions.\ng(x) = 2x^3 - 8x^2 + 5\nf(x) = (2x^3 - 8x^2 + 5)^6\ng(x) =…

Explanation:

Step1: Identify the outer - inner functions for $f(x)$

Let $u = 2x^{3}-8x^{2}+5$, so $f(x)=u^{6}$.

Step2: Find the derivative of the outer function

The derivative of $y = u^{6}$ with respect to $u$ is $\frac{dy}{du}=6u^{5}$.

Step3: Find the derivative of the inner function

We know that $u = 2x^{3}-8x^{2}+5$, and its derivative $\frac{du}{dx}=6x^{2}-16x$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $u = 2x^{3}-8x^{2}+5$ back in, we get $f^{\prime}(x)=6(2x^{3}-8x^{2}+5)^{5}(6x^{2}-16x)$.

Answer:

$f^{\prime}(x)=6(2x^{3}-8x^{2}+5)^{5}(6x^{2}-16x)$