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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{- 1}^{3} x^{2} e^{x^{3} + 1} d x

\int cot (14 x) d x

\int sec^{2} (2 y) d y

\frac{d}{d x} (4 (x + 2)^{\frac{3}{2}})

\frac{d y}{d t} = - \frac{y}{t} - 6

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