find the derivative of the function

Question

g(x) = (4x^2 + 3x +4) e^x

g'(x) = ?? leave answer in ( 4x^2 + ?x + ?)

Answer 1

There are going to be 3 Derivative Definitions that you'll need to have for this problem:

d/dx (f(x) * g(x)) = f'(x)g(x) + f(x)g'(x) This is the Product Rule

d/dx (A * x^B) = AB * x^(B-1) This is the Power Rule

d/dx (e^Cx) = C e^Cx This is the Chain Rule infused with the Exponential Function, e^x

d/dx( ( 4x^2 + 3x + 4 ) e^x ) = d/dx( 4x^2 + 3x + 4 ) * e^x + ( 4x^2 + 3x + 4 ) * d/dx( e^x )

d/dx( ( 4x^2 + 3x + 4 ) e^x ) = ( 8x + 3 + 0 ) e^x + ( 4x^2 + 3x + 4 ) * e^x

∴ d/dx( ( 4x^2 + 3x + 4 ) e^x ) = ( 4x^2 + 11x + 7 ) e^x

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Answer 2

g(x) = e^x(4x² + 3x + 4)

Using the Product Rule, g'(x) = e^x(4x² + 3x + 4) + e^x(8x + 3) = e^x(4x² + 11x + 7)