question 5 of 16. step 1 of 1 correct find the derivative for the given function. write your answer using…

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt[5]{8x^{3}-3}$ as $y=(8x^{3}-3)^{\frac{1}{5}}$ using the rule $\sqrt[n]{a}=a^{\frac{1}{n}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{5}}$ and $u = 8x^{3}-3$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: $\frac{dy}{du}=\frac{1}{5}u^{-\frac{4}{5}}$ (using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$ with $n=\frac{1}{5}$). Then find $\frac{du}{dx}$: $\frac{du}{dx}=24x^{2}$ (using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$ for $a = 8,n = 3$ and $\frac{d}{dx}(c)=0$ for $c=-3$).

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = 8x^{3}-3$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$: $\frac{dy}{dx}=\frac{1}{5}(8x^{3}-3)^{-\frac{4}{5}}\cdot24x^{2}=\frac{24x^{2}}{5(8x^{3}-3)^{\frac{4}{5}}}$.

Answer:

$\frac{24x^{2}}{5(8x^{3}-3)^{\frac{4}{5}}}$