What is (\partial)/(\partial x)(xe^{s/x}) ?

The answer to (\partial)/(\partial x)(xe^{s/x}) is e^{s/x}-(e^{s/x}s)/x

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(\partial)/(\partial x)(xe^{s/x})

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

e^{\frac{s}{x}} - \frac{e ^{\frac{s}{x}} s}{x}

Solution steps

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

Treat

s

as a constant

Apply the Product Rule:

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

f = x, g = e^{\frac{s}{x}}

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

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

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

= 1 \cdot e^{\frac{s}{x}} + (- \frac{s e ^{\frac{s}{x}}}{x ^{2}}) x

Simplify

1 \cdot e^{\frac{s}{x}} + (- \frac{s e ^{\frac{s}{x}}}{x ^{2}}) x : e^{\frac{s}{x}} - \frac{e ^{\frac{s}{x}} s}{x}

= e^{\frac{s}{x}} - \frac{e ^{\frac{s}{x}} s}{x}

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

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

y = 2 sin (x), at x = \frac{3}{4} π

h (x) = \frac{3 x ^{3} - 3}{ln ( x )}

f (x) = 1 - 2 x + x^{2}

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