What is (\partial)/(\partial x)(x-1/x e^{y/x}) ?

The answer to (\partial)/(\partial x)(x-1/x e^{y/x}) is 1-(-ye^{y/x}-e^{y/x}x)/(x^3)

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(\partial)/(\partial x)(x-1/x e^{y/x})

\frac{\partial}{\partial x} (x - \frac{1}{x} e^{\frac{y}{x}})

1 - \frac{- y e ^{\frac{y}{x}} - e ^{\frac{y}{x}} x}{x ^{3}}

Solution steps

\frac{\partial}{\partial x} (x - \frac{1}{x} e^{\frac{y}{x}})

Simplify

x - \frac{1}{x} e^{\frac{y}{x}} : x - \frac{e ^{\frac{y}{x}}}{x}

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

Treat

y

as a constant

Apply the Sum/Difference Rule:

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

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

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

\frac{\partial}{\partial x} (\frac{e ^{\frac{y}{x}}}{x}) = \frac{- y e ^{\frac{y}{x}} - e ^{\frac{y}{x}} x}{x ^{3}}

= 1 - \frac{- y e ^{\frac{y}{x}} - e ^{\frac{y}{x}} x}{x ^{3}}

\frac{d}{d s} (\frac{2}{s + s ^{2} + 4})

\frac{\partial}{\partial x} (6 x y - 12 x)

(2 x + 4) d x + (3 y - 1) d y = 0

x \to 0 lim (\frac{1}{x + 3})

\int cos^{4} (2 x) sin (2 x) d x

What is (\partial)/(\partial x)(x-1/x e^{y/x}) ?
The answer to (\partial)/(\partial x)(x-1/x e^{y/x}) is 1-(-ye^{y/x}-e^{y/x}x)/(x^3)