What is (\partial)/(\partial y)(sqrt(xye^y)) ?

The answer to (\partial)/(\partial y)(sqrt(xye^y)) is (e^ysqrt(x)(1+y))/(2sqrt(y)\sqrt{e^y)}

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(\partial)/(\partial y)(sqrt(xye^y))

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

\frac{1}{2 x y e ^{y}} \frac{\partial}{\partial y} (x y e^{y})

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

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

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

Simplify

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

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

\int \frac{2 y}{1 - y ^{2}} d y

g (x) = x e^{x}

\int \frac{4}{x ^{2}} - \frac{7}{x ^{5}} d x

f (x) = x^{3} + 7 x - 7, (1, 1)

y^{^{''''}} = 0

What is (\partial)/(\partial y)(sqrt(xye^y)) ?
The answer to (\partial)/(\partial y)(sqrt(xye^y)) is (e^ysqrt(x)(1+y))/(2sqrt(y)\sqrt{e^y)}