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

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

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(\partial)/(\partial x)(-xe^{y/x})

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

- e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= - e^{\frac{y}{x}} + \frac{e ^{\frac{y}{x}} y}{x}

t^{3} \frac{d y}{d t} + 3 t^{2} y = 9 cos (t), y (π) = 0

cos (10 t)

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

\int (x - 2) (2 x + 3) d x

x \frac{d y}{d x} = y ln (x y)

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