Consider the following function: g(x) = 8x²eˣ Step 2 of 2: Find the second derivative of the above function. g''(x) =

To find the second derivative of the function g ( x ) = 8 x 2 e x , we will go through the following steps:
Step 1: Find the first derivative, g ′ ( x ) .
g ( x ) = 8 x 2 e x is a product of two functions, u ( x ) = 8 x 2 and v ( x ) = e x . We'll use the product rule, which states that if u ( x ) and v ( x ) are differentiable functions, then:
( uv ) ′ = u ′ v + u v ′
First, find the derivatives of u ( x ) and v ( x ) :

u ′ ( x ) = d x d ​ ( 8 x 2 ) = 16 x
v ′ ( x ) = d x d ​ ( e x ) = e x

Applying the product rule:
g ′ ( x ) = u ′ v + u v ′ = ( 16 x ) ( e x ) + ( 8 x 2 ) ( e x ) = 16 x e x + 8 x 2 e x
We can simplify this to:
g ′ ( x ) = e x ( 16 x + 8 x 2 )
Step 2: Find the second derivative, g ′′ ( x ) .
To find g ′′ ( x ) , we differentiate g ′ ( x ) :
g ′ ( x ) = e x ( 16 x + 8 x 2 )
This again involves using the product rule. Let u ( x ) = e x and v ( x ) = 16 x + 8 x 2 .
Find the derivatives:

u ′ ( x ) = e x
v ′ ( x ) = d x d ​ ( 16 x + 8 x 2 ) = 16 + 16 x

Now, apply the product rule:
g ′′ ( x ) = u ′ v + u v ′ = ( e x ) ( 16 x + 8 x 2 ) + ( e x ) ( 16 + 16 x )
Combine terms:
g ′′ ( x ) = e x ( 16 x + 8 x 2 ) + e x ( 16 + 16 x )
g ′′ ( x ) = e x ( 16 x + 8 x 2 + 16 + 16 x )
Combine like terms:
g ′′ ( x ) = e x ( 8 x 2 + 32 x + 16 )
Thus, the second derivative of the function g ( x ) = 8 x 2 e x is:
g ′′ ( x ) = e x ( 8 x 2 + 32 x + 16 )

Answered by MasonWilliamTurner | 2025-07-22