Question

F(x) = ( - 3x² + 2 – 2)e". Find f'(x) and f''(x).
f'(2) =
f''(c)
=

Answer 1

-3x²-5xeˣ-eˣ

-3eˣx²-11eˣx-6eˣ

Explanation:

I'm going to go by the picture and not what you wrote in your title.

To find the derivative of this we have to apply the product rule

(a*b)'=

a'*b+a*b'

We plug in our numbers and get

(-3x²+x-2)'*eˣ+(-3x²+x-2)*eˣ'

Now we can evaluate the derivatives and simplify

(-3x²+x-2)'= -6x+1

eˣ'=eˣ

which means we have

(-6x+1)*eˣ+(-3x²+x-2)*eˣ

Simplify

-6xeˣ+eˣ-3x²eˣ+xeˣ-2eˣ

Combine like terms

-3x²eˣ-5xeˣ-eˣ

Now we just need to find the derivative of this

We can apply the same product rule as we did before

(-3x²eˣ)'

Let's start by factoring out the -3 to get

-3(x²eˣ)'

which is equal to

-3(x²eˣ'+x²'eˣ)

Compute this and get

-3(x²eˣ+2xeˣ)= -3x²eˣ-6xeˣ

Now let's find the derivative of the second part

(-5xeˣ)'

-5(x'eˣ+xeˣ')

-5(eˣ+xeˣ)

-5eˣ-5xeˣ

Which means we have

(-3x²eˣ-6xeˣ)+(-5eˣ-5xeˣ)-eˣ

Combine like terms and get

-3eˣx²-11eˣx-6eˣ