What is the derivative of sqrt(x)*e^{-x} ?

The derivative of sqrt(x)*e^{-x} is (e^{-x})/(2sqrt(x))-e^{-x}sqrt(x)

What is the first derivative of sqrt(x)*e^{-x} ?

The first derivative of sqrt(x)*e^{-x} is (e^{-x})/(2sqrt(x))-e^{-x}sqrt(x)

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\frac{d}{d x} (x \cdot e^{- x})

\frac{e ^{- x}}{2 x} - e^{- x} x

Solution steps

\frac{d}{d x} (x e^{- x})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{- x}

= \frac{d}{d x} (x) e^{- x} + \frac{d}{d x} (e^{- x}) x

\frac{d}{d x} (x) = \frac{1}{2 x}

\frac{d}{d x} (e^{- x}) = - e^{- x}

= \frac{1}{2 x} e^{- x} + (- e^{- x}) x

Simplify

\frac{1}{2 x} e^{- x} + (- e^{- x}) x : \frac{e ^{- x}}{2 x} - e^{- x} x

= \frac{e ^{- x}}{2 x} - e^{- x} x

What is the derivative of sqrt(x)*e^{-x} ?
The derivative of sqrt(x)*e^{-x} is (e^{-x})/(2sqrt(x))-e^{-x}sqrt(x)
What is the first derivative of sqrt(x)*e^{-x} ?
The first derivative of sqrt(x)*e^{-x} is (e^{-x})/(2sqrt(x))-e^{-x}sqrt(x)