What is (\partial)/(\partial x)(y(e^x+xe^x)) ?

The answer to (\partial)/(\partial x)(y(e^x+xe^x)) is y(2e^x+e^xx)

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(\partial)/(\partial x)(y(e^x+xe^x))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

Simplify

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

g (x) = \frac{1}{4 x ^{2} + 2 x}

2 t + 3 t \cdot cos (2 t)

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

\int 2 x \cdot sin (x) e^{x} d x

t \cdot t y^{^{''}} + 5 t y^{^{'}} + 4 y = 0

What is (\partial)/(\partial x)(y(e^x+xe^x)) ?
The answer to (\partial)/(\partial x)(y(e^x+xe^x)) is y(2e^x+e^xx)