Question

How do you differentiate `g(x) = (2x^2 + 4e^x) ( 2x + 2)` using the product rule?

Answer 1

Use expansion, apply linearity and product rule to get:
`g'(x) =4(2(x+2)e^x+x(3x+2))`

Explanation:

The approach that I am going to use to differentiate `g(x) = (2x^2 + 4e^x) ( 2x + 2)` is expansion:
`g(x) = (2x^3 + 8xe^x) + ( 4x^2 + 8e^x)`
Now we can apply linearity and write:
`g'(x) =2(d(x^3))/dx + (d(color(blue)(8xe^x)))/dx+4(d(x^2))/dx + 8(d(e^x))/dx` Now you can differentiate each term in straight forward manner except for `color(blue)(blue)`. For `color(blue)(blue)` you can use product rule: `color(blue)(blue' = 8e^x + 8xe^x)`
Putting it all together you have:
`g'(x) =4(2(x+2)e^x+x(3x+2))`