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

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

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

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

2 e^{- x} x - e^{- x} (x^{2} + y^{2})

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

x \to 1 lim (\frac{4}{x ^{2} - 1})

\frac{d}{d x} ((3 x + 1)^{4})

\frac{d}{d x} (cos (y) \frac{d y}{d x})

5 x^{- 4}

\int (x + 1) \cdot (x - x + 1) d x

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