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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{- 1}^{1} x cos (nπ x) d x

\int \frac{1}{1 - v ^{2}} d v

\int (x - 2)^{2} d x

\frac{d}{d x} (\frac{1 - ln ( x )}{1 + ln ( x )})

\frac{5}{x}, at x = a

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