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

1 Answer
Mar 9, 2016

Answer:

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)) #