What is (\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2)))) ?

The answer to (\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2)))) is -y/(x^2+y^2)

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(\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2))))

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (5 x + \frac{400}{x})

7 y^{^{'}} = y^{2}

f (x) = x e^{3 x^{2} + 1}

\int \frac{1}{( y + 1 ) ( y - 2 )} d y

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

What is (\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2)))) ?
The answer to (\partial)/(\partial x)(arccos(x/(sqrt(x^2+y^2)))) is -y/(x^2+y^2)