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

The answer to (\partial)/(\partial y)((x/y)e^{xy}) is x(-(e^{xy}}{y^2}+\frac{e^{xy}x)/y)

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(\partial)/(\partial y)((x/y)e^{xy})

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = \frac{1}{y}, g = e^{x y}

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

\frac{\partial}{\partial y} (\frac{1}{y}) = - \frac{1}{y ^{2}}

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

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

Simplify

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

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

\frac{\partial}{\partial z} (e^{x} sin (y))

y^{^{''}} - 10 y^{^{'}} + 25 y = t^{- 8} e^{5 t}

\int 7 x e^{5 x} d x

\int (2 - 3 x) cos (4 x) d x

\frac{\partial}{\partial x} (x^{2} + y^{2})

What is (\partial)/(\partial y)((x/y)e^{xy}) ?
The answer to (\partial)/(\partial y)((x/y)e^{xy}) is x(-(e^{xy}}{y^2}+\frac{e^{xy}x)/y)