What is (\partial)/(\partial x)(y^3e^{(x^2)/(2y)}) ?

The answer to (\partial)/(\partial x)(y^3e^{(x^2)/(2y)}) is e^{(x^2)/(2y)}y^2x

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(\partial)/(\partial x)(y^3e^{(x^2)/(2y)})

\frac{\partial}{\partial x} (y^{3} e^{\frac{x ^{2}}{2 y}})

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

Solution steps

\frac{\partial}{\partial x} (y^{3} e^{\frac{x ^{2}}{2 y}})

Treat

y

as a constant

Take the constant out:

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

= y^{3} \frac{\partial}{\partial x} (e^{\frac{x ^{2}}{2 y}})

Apply the chain rule:

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

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

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

= y^{3} e^{\frac{x ^{2}}{2 y}} \frac{x}{y}

Simplify

y^{3} e^{\frac{x ^{2}}{2 y}} \frac{x}{y} : e^{\frac{x ^{2}}{2 y}} y^{2} x

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

\frac{f ( x )}{g ( x )}

\int e^{5 x} 3 e^{5 x} + 2 d x

\int \frac{3 x - 1}{4 - a x} d x

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

y^{^{''}} + 11 y^{^{'}} + 18 y = 0

What is (\partial)/(\partial x)(y^3e^{(x^2)/(2y)}) ?
The answer to (\partial)/(\partial x)(y^3e^{(x^2)/(2y)}) is e^{(x^2)/(2y)}y^2x