Find the derivative of the function. $f ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }$
A) $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$
B) $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 + 34 x ) } { 2 \sqrt { 2 - 2 x } }$
C) $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 - 34 x ) } { 2 \sqrt { 2 - 2 x } }$
D) $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 32 - 2 x ) } { 2 \sqrt { 2 - 2 x } }$
E) $f ^ { \prime } ( x ) = \frac { x ^ { 7 } ( 2 + 2 x ) } { 2 \sqrt { 2 - 2 x } }$

Question

Quizplus · Accepted Answer

To find the derivative of $f(x) = x^8\sqrt{2 - 2x}$, we use the product rule and the chain rule. The derivative of $x^8$ is $8x^7$, and the derivative of $\sqrt{2 - 2x}$ is $\frac{-2}{2\sqrt{2 - 2x}} = \frac{-1}{\sqrt{2 - 2x}}$. Applying the product rule, we get:$$f'(x) = 8x^7\sqrt{2 - 2x} + x^8\left(\frac{-1}{\sqrt{2 - 2x}}\right)(-2) = \frac{8x^7(2 - 2x) + 2x^8}{2\sqrt{2 - 2x}} = \frac{x^7(16 - 16x + 2x)}{2\sqrt{2 - 2x}} = \frac{x^7(32 - 34x)}{2\sqrt{2 - 2x}}.$$

Find the Derivative of the Function f(x)=x82−2xf ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }f(x)=x82−2x​

Find the Derivative of the Function $f ( x ) = x ^ { 8 } \sqrt { 2 - 2 x }$