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

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} (- \frac{8}{1 - 8 x})

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

\int_{0}^{1} \frac{1}{x + 1} d x

\int_{0}^{1} (x + x) d x

\int \frac{( 6 - x )}{4 x ^{2} - 12 x + 7} d x

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))