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

The answer to (\partial)/(\partial x)((2y)/(x^2-y^2)) is -(4yx)/((x^2-y^2)^2)

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

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

- \frac{4 y x}{( x ^{2} - y ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

= - \frac{4 y x}{( x ^{2} - y ^{2} ) ^{2}}

y^{^{''}} + 10 y^{^{'}} + 25 y = 14 e^{- 5 x}

x \to \infty lim (- \frac{1}{x})

x \to 1 lim (cos (\frac{π}{3}))

\frac{d}{d x} (\frac{2}{3} π + 1)

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

What is (\partial)/(\partial x)((2y)/(x^2-y^2)) ?
The answer to (\partial)/(\partial x)((2y)/(x^2-y^2)) is -(4yx)/((x^2-y^2)^2)