What is (\partial)/(\partial x)(arctan(z/x)) ?

The answer to (\partial)/(\partial x)(arctan(z/x)) is -z/(z^2+x^2)

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(\partial)/(\partial x)(arctan(z/x))

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

- \frac{z}{z ^{2} + x ^{2}}

Solution steps

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

Treat

z

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (\frac{z}{x}) = - \frac{z}{x ^{2}}

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

Simplify

\frac{1}{( \frac{z}{x} ) ^{2} + 1} (- \frac{z}{x ^{2}}) : - \frac{z}{z ^{2} + x ^{2}}

= - \frac{z}{z ^{2} + x ^{2}}

y^{^{''}} - 10 y^{^{'}} + 50 y = 0, y (0) = 5, y^{^{'}} (0) = - 2

5 y^{^{''}} + 30 y = 0, y (0) = 5, y^{^{'}} (0) = 5

\int \frac{1}{a ^{2} + u ^{2}} d u

\frac{\partial}{\partial y} ((2 x - 5 y)^{2})

(10 y^{2} - x y) d x + (x^{2}) d y = 0

What is (\partial)/(\partial x)(arctan(z/x)) ?
The answer to (\partial)/(\partial x)(arctan(z/x)) is -z/(z^2+x^2)