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

The derivative of x(e^{-x}) is e^{-x}-e^{-x}x

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

The first derivative of x(e^{-x}) is e^{-x}-e^{-x}x

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

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

\frac{d x}{d x} = 1

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

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

Simplify

= e^{- x} - e^{- x} x

\frac{\partial}{\partial x} (\frac{x y}{e ^{x^{2} + y^{2}}})

ln (arctan (2 x^{3}))

\int \frac{3}{x} + 2 e^{x} d x

\frac{d}{d x} (3 x - x^{2})

\frac{d}{d x} (\frac{4 - x ^{2}}{4 + x ^{2}})

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