What is (\partial)/(\partial x)(sqrt(x)e^x) ?

The answer to (\partial)/(\partial x)(sqrt(x)e^x) is (e^x)/(2sqrt(x))+e^xsqrt(x)

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(\partial)/(\partial x)(sqrt(x)e^x)

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = e^{x}

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

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

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

= \frac{1}{2 x} e^{x} + e^{x} x

\frac{1}{2 x} e^{x} = \frac{e ^{x}}{2 x}

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

\frac{d}{d x} (2 x^{2} e^{\frac{x}{2}})

\frac{( s - 2 )}{s ^{2} - 2 s + 10}

x \to - 5 lim (\frac{1}{1 + 5})

\frac{d}{d x} ((ln (x))^{2} sin (x))

\frac{d y}{d x} = (x - 3) (y + 1)^{\frac{2}{3}}

What is (\partial)/(\partial x)(sqrt(x)e^x) ?
The answer to (\partial)/(\partial x)(sqrt(x)e^x) is (e^x)/(2sqrt(x))+e^xsqrt(x)