What is (\partial)/(\partial y)(arctan(xy^2)) ?

The answer to (\partial)/(\partial y)(arctan(xy^2)) is (2xy)/(x^2y^4+1)

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(\partial)/(\partial y)(arctan(xy^2))

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

\frac{2 x y}{x ^{2} y ^{4} + 1}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{( x y ^{2} ) ^{2} + 1} \cdot 2 x y : \frac{2 x y}{x ^{2} y ^{4} + 1}

= \frac{2 x y}{x ^{2} y ^{4} + 1}

\int 2 t e^{2 t} d t

x \to 0 - lim (\frac{3}{e ^{x} + 1})

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

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

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

What is (\partial)/(\partial y)(arctan(xy^2)) ?
The answer to (\partial)/(\partial y)(arctan(xy^2)) is (2xy)/(x^2y^4+1)