part 1 of 1 - question 50 of 50 2 points find the derivative. y = ∛(-5r² + 3) a. dy/dr = -10r /…

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt[3]{- 5r^{2}+3}=(-5r^{2}+3)^{\frac{1}{3}}$ using the rule $\sqrt[n]{a}=a^{\frac{1}{n}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(r))$, then $\frac{dy}{dr}=f^{\prime}(g(r))\cdot g^{\prime}(r)$. Let $u=-5r^{2}+3$, so $y = u^{\frac{1}{3}}$. First, find $\frac{dy}{du}$ and $\frac{du}{dr}$. $\frac{dy}{du}=\frac{1}{3}u^{-\frac{2}{3}}$ (using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$ with $n=\frac{1}{3}$), and $\frac{du}{dr}=-10r$ (using the power rule $\frac{d}{dr}(ar^{n})=nar^{n - 1}$ for $a=-5$ and $n = 2$).

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

By the chain - rule $\frac{dy}{dr}=\frac{dy}{du}\cdot\frac{du}{dr}$. Substitute $u=-5r^{2}+3$ back into $\frac{dy}{du}$: $\frac{dy}{dr}=\frac{1}{3}(-5r^{2}+3)^{-\frac{2}{3}}\cdot(-10r)=-\frac{10r}{3(\sqrt[3]{-5r^{2}+3})^{2}}$.

Answer:

A. $\frac{dy}{dr}=-\frac{10r}{3(\sqrt[3]{-5r^{2}+3})^{2}}$