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

(Question Restore: portions of this question have been edited or deleted!)

Answer 1

Emily Ammerman

(D) is an intermediate step.

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

The Product Rule for Differentiation will be used:

# 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) #

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Answer 2

Paisley Johnson

The derivative of ( (x^2+2)e^{4x} ) is ( (2x + 4)e^{4x} + (x^2+2)4e^{4x} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.