How do you find the derivative of $(x^{\frac{1}{2}}) (e^{- x})$ $(x^(1/2))(e^(-x))$ ?

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Answer 1 · 2016-10-02T11:18:13

$e^{- x} [\frac{1 - 2 x}{2 \sqrt{x}}]$ $e^{-x} [ (1-2x)/(2sqrt(x)) ]$

The rule for deriving a product is

$(f*g)' = f'g+fg'$

In your case, we have

$f(x) = x^{1/2} = sqrt(x)$

$f'(x) = 1/2 x^{-1/2} = 1/(2sqrt(x))$

$g(x) = e^{-x}$

$g'(x) = e^{-x} * d/dx (-x) = e^{-x}(-1) = -e^{-x}$

I used the power rule $d/dx x^n = nx^{n-1}$ for $f'(x)$ and the composite rule $d/dx f(g(x)) = f'(g(x))*g'(x)$ for g'(x)#

Now that we have all the elements, let's plug everything into the formula:

$f'g+fg' = 1/(2sqrt(x))*e^{-x}+ sqrt(x)(-e^{-x})$

Which we can rearrange factoring $e^{-x}$ :

$e^{-x} [1/(2sqrt(x))-sqrt(x)]$

and again, if you prefer,

$e^{-x} [ (1-2x)/(2sqrt(x)) ]$

How do you find the derivative of (x^(1/2))(e^(-x)) (x12)(e−x) (x^(1/2))(e^(-x)) ?