Differentiate the function.
$$y = x \left( 5 ^ { 2 x } \right)$$
A) $( 2 \ln 5 ) \left( 5 ^ { 2 x } \right) x$
B) $[ 1 + ( 2 \ln 5 ) x ] 5 ^ { 2 x }$
C) $[ 1 + ( 5 \ln 2 ) x ] 5 ^ { 2 x }$
D) $( 2 \ln 5 ) 5 ^ { 2 x }$

Question

Quizplus · Accepted Answer

This problem requires the product rule, which states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

Let u = x and v = 5^(2x).  Then we have y = u*v.

The derivative of v = 5^(2x) is:  
v' = 5^(2x) * ln(5) * 2 = 2ln(5) * 5^(2x)

The derivative of u = x is:
u' = 1

Then, using the product rule, we have:

y' = u' * v + u * v'

y' = 1 * 5^(2x)*x + x * 2ln(5) * 5^(2x)

y' = [1 + 2ln(5)x]5^(2x)

Therefore, the answer is B: [1 + (2ln(5))x]5^(2x).

Differentiate the Function y=x(52x)y = x \left( 5 ^ { 2 x } \right)y=x(52x)

Differentiate the Function $y = x \left( 5 ^ { 2 x } \right)$