`y = sqrt(1 + x e^(-2x))` Find the derivative of the function.

Textbook Question

Chapter 3, 3.4 - Problem 36 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`y=sqrt(1+xe^(-2x))`

`y'=(1/2)*((1+xe^(-2x))^((1/2)-1)) *d/dx (sqrt(1+xe^(-2x)))`

`y'=(1/(2sqrt(1+xe^(-2x)))) *(xd/dx e^(-2x) +e^(-2x)d/dx x)`

`y'=(1/(2sqrt(1+xe^(-2x)))) *(xe^(-2x)*(-2) + e^(-2x))`

`y'=(e^(-2x) *(1-2x))/(2sqrt(1+xe^(-2x)))`

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hkj1385 | (Level 1) Assistant Educator

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Note:- 1) if y = e^(ax) ; then dy/dx = a*e^(ax)

2) if y = sqrt(ax) ; then dy/dx = [1/sqrt(ax)]*a

Thus, 

y = sqrt[1 + {x*e^(-2x)}] 

dy/dx = y' = [1/2 sqrt[1 + {x*e^(-2x)}]]*[e^(-2x) - {2*x*e^(-2x)}]

or, dy/dx = y' = [(e^-2x)*(1-2x)]/2 sqrt[1 + {x*e^(-2x)}]

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