Find $f ^ { \prime }$ in terms of $g ^ { \prime }$.
$$f ( x ) = x ^ { 5 } g ( x )$$
Select the correct answer.
A) $f ^ { \prime } ( x ) = 5 x ^ { 4 } + g ^ { \prime } ( x )$
B) $f ^ { \prime } ( x ) = x ^ { 5 } g ( x ) + 5 x ^ { 5 } g ^ { \prime } ( x )$
C) $f ^ { \prime } ( x ) = 5 x ^ { 4 } g ( x ) + x ^ { 5 } g ^ { \prime } ( x )$
D) $f ^ { \prime } ( x ) = 5 x ^ { 4 } g ^ { \prime } ( x )$
E) $f ^ { \prime } ( x ) = 5 x f ^ { \prime } ( x ) + 5 x g ^ { \prime } ( x )$

Question

Quizplus · Accepted Answer

To find the derivative of the function, we use the product rule, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the derivative of the second function times the first function. In this case, we have:

$f(x) = x^5 g(x)$

So, using the product rule:

$f'(x) = (5x^4g(x)) + (x^5g'(x))$

Simplifying:

$f'(x) = 5x^4g(x) + x^5g'(x)$

Therefore, the correct answer is C.

Find f′f ^ { \prime }f′ In Terms Of g′g ^ { \prime }g′

Find $f ^ { \prime }$ In Terms Of $g ^ { \prime }$