What is (\partial)/(\partial x)(arctan(x^4y^3)) ?

The answer to (\partial)/(\partial x)(arctan(x^4y^3)) is (4x^3y^3)/(x^8y^6+1)

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(\partial)/(\partial x)(arctan(x^4y^3))

\frac{\partial}{\partial x} (arctan (x^{4} y^{3}))

\frac{4 x ^{3} y ^{3}}{x ^{8} y ^{6} + 1}

Solution steps

\frac{\partial}{\partial x} (arctan (x^{4} y^{3}))

Treat

y

as a constant

Apply the chain rule:

\frac{1}{( x ^{4} y ^{3} ) ^{2} + 1} \frac{\partial}{\partial x} (x^{4} y^{3})

= \frac{1}{( x ^{4} y ^{3} ) ^{2} + 1} \frac{\partial}{\partial x} (x^{4} y^{3})

\frac{\partial}{\partial x} (x^{4} y^{3}) = 4 y^{3} x^{3}

= \frac{1}{( x ^{4} y ^{3} ) ^{2} + 1} \cdot 4 y^{3} x^{3}

Simplify

\frac{1}{( x ^{4} y ^{3} ) ^{2} + 1} \cdot 4 y^{3} x^{3} : \frac{4 x ^{3} y ^{3}}{x ^{8} y ^{6} + 1}

= \frac{4 x ^{3} y ^{3}}{x ^{8} y ^{6} + 1}

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What is (\partial)/(\partial x)(arctan(x^4y^3)) ?
The answer to (\partial)/(\partial x)(arctan(x^4y^3)) is (4x^3y^3)/(x^8y^6+1)