$\text { Evaluate the derivative of the function } y = \sqrt [ 5 ] { 2 x ^ { 5 } + 2 x } \text { at the point } x = 3 \text {. }$
A)$y ^ { t } ( 3 ) = \frac { 812 } { ( 492 ) ^ { \frac { 3 } { 5 } } }$
B) $y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 3 } { 5 } } }$
C) $y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 4 } { 5 } } }$
D)$y ^ { t } ( 3 ) = \frac { 812 } { 407 ( 492 ) ^ { \frac { 4 } { 5 } } }$
E)$y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 6 } { 5 } } }$

Question

$\text { Evaluate the derivative of the function } y = \sqrt [ 5 ] { 2 x ^ { 5 } + 2 x } \text { at the point } x = 3 \text {. }$ 
A)$y ^ { t } ( 3 ) = \frac { 812 } { ( 492 ) ^ { \frac { 3 } { 5 } } }$
B) $y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 3 } { 5 } } }$
C) $y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 4 } { 5 } } }$
D)$y ^ { t } ( 3 ) = \frac { 812 } { 407 ( 492 ) ^ { \frac { 4 } { 5 } } }$
E)$y ^ { t } ( 3 ) = \frac { 812 } { 5 ( 492 ) ^ { \frac { 6 } { 5 } } }$

Quizplus · Accepted Answer

To find the derivative of $y = \sqrt[5]{2x^5 + 2x}$, we use the chain rule. The outer function is $u^{1/5}$ and the inner function is $u = 2x^5 + 2x$. The derivative of the outer function with respect to $u$ is $\frac{1}{5}u^{-4/5}$, and the derivative of the inner function with respect to $x$ is $10x^4 + 2$. Thus, the derivative of $y$ with respect to $x$ is $\frac{1}{5}(2x^5 + 2x)^{-4/5}(10x^4 + 2)$. Evaluating this at $x = 3$, we get $\frac{1}{5}(2(3)^5 + 2(3))^{-4/5}(10(3)^4 + 2) = \frac{1}{5}(486 + 6)^{-4/5}(810 + 2) = \frac{812}{5(492)^{4/5}}$, which matches choice C.

Evaluate the derivative of the function y=2x5+2x5 at the point x=3. \text { Evaluate the derivative of the function } y = \sqrt [ 5 ] { 2 x ^ { 5 } + 2 x } \text { at the point } x = 3 \text {. } Evaluate the derivative of the function y=52x5+2x​ at the point x=3.

$\text { Evaluate the derivative of the function } y = \sqrt [ 5 ] { 2 x ^ { 5 } + 2 x } \text { at the point } x = 3 \text {. }$