What is the derivative of? : #(x^2+2)e^(4x) #

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Answer 1 · 2018-01-15T16:40:59

(D) is an intermediate step.

# dy/dx = 2(2x^2+x+4)e^(4x) #

We will apply the Product Rule for Differentiation:

# d/dx(uv)=u(dv)/dx+(du)/dxv #, or, # (uv)' = (du)v + u(dv) #

So with # y = (x^2+2)e^(4x) #;

# { ("Let", u = x^2+2, => (du)/dx = 2x), ("And" ,v = e^(4x), =>(dv)/dx = 4e^(4x) ) :}#

Then:

# d/dx(uv)=u(dv)/dx + (du)/dxv #

Giving:

# dy/dx = (x^2+2)(4e^(4x)) + (2x)(e^(4x)) #
# \ \ \ \ \ \ = 2(2x^2+x+4)e^(4x) #