What is (\partial)/(\partial x)(x^2e^{-x/y}) ?

The answer to (\partial)/(\partial x)(x^2e^{-x/y}) is 2e^{-x/y}x-(e^{-x/y}x^2)/y

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(\partial)/(\partial x)(x^2e^{-x/y})

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

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

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

f = x^{2}, g = e^{- \frac{x}{y}}

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

\frac{\partial}{\partial x} (e^{- \frac{x}{y}}) = - \frac{e ^{- \frac{x}{y}}}{y}

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

Simplify

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

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

\int_{1}^{5} \frac{e ^{\frac{x}{2}}}{x ^{3}} d x

3 e^{x}, 3 x e^{x^{2}}, [0, 1]

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\int \frac{( x + 6 )}{( x + 1 )} d x

\int_{2}^{3} \frac{ln ( 2 x ^{5} )}{x ^{2}} d x

What is (\partial)/(\partial x)(x^2e^{-x/y}) ?
The answer to (\partial)/(\partial x)(x^2e^{-x/y}) is 2e^{-x/y}x-(e^{-x/y}x^2)/y