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

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

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(\partial)/(\partial x)(-x/(sqrt(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}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

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

\frac{d}{d x} ((2 x)^{x})

y^{^{'''}} - 8 y = 0

\frac{d}{d x} (ln (2 - 5 x))

\frac{d}{d x} (arctan (x) - x^{2})

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