What is (\partial)/(\partial y)(arctan(xsqrt(y))) ?

The answer to (\partial)/(\partial y)(arctan(xsqrt(y))) is x/(2sqrt(y)(x^2y+1))

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(\partial)/(\partial y)(arctan(xsqrt(y)))

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{a x + b y ( x )}{c x + d y})

\frac{d}{d x} (\frac{e ^{x} - 1}{e ^{x} + 1})

\frac{d}{d x} (\frac{1}{( x ^{4} + 1 ) ^{3}})

4.9 x^{1.9}, at x = 1.4

\frac{d}{d x} (x^{a} ln (x))

What is (\partial)/(\partial y)(arctan(xsqrt(y))) ?
The answer to (\partial)/(\partial y)(arctan(xsqrt(y))) is x/(2sqrt(y)(x^2y+1))