What is (\partial)/(\partial x)(sqrt(x/(x+y))) ?

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

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(\partial)/(\partial x)(sqrt(x/(x+y)))

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

\frac{y}{2 x ( x + y ) ^{\frac{3}{2}}}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

\frac{1}{2 \frac{x}{x + y}} \frac{\partial}{\partial x} (\frac{x}{x + y})

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

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

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

Simplify

\frac{1}{2 \frac{x}{x + y}} \cdot \frac{y}{( x + y ) ^{2}} : \frac{y}{2 x ( x + y ) ^{\frac{3}{2}}}

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

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

\frac{d}{d x} (ln (x + x^{2} + y^{2}))

\int \frac{x ^{6}}{x ^{7} - 2} d x

\int_{0}^{2} ((- x + 5) - (x^{2} - 9)) d x

\frac{d}{d x} (\frac{2 a}{x ^{3}})

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