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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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