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

The answer to (\partial)/(\partial x)(-y/(sqrt(y^2-x^2))) is -(yx)/((y^2-x^2)^{3/2)}

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

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

- \frac{y x}{( y ^{2} - x ^{2} ) ^{\frac{3}{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})^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( y ^{2} - x ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (y^{2} - x^{2})

= - \frac{1}{2 ( y ^{2} - x ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (y^{2} - x^{2})

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

= - y (- \frac{1}{2 ( y ^{2} - x ^{2} ) ^{\frac{3}{2}}} (- 2 x))

Simplify

- y (- \frac{1}{2 ( y ^{2} - x ^{2} ) ^{\frac{3}{2}}} (- 2 x)) : - \frac{y x}{( y ^{2} - x ^{2} ) ^{\frac{3}{2}}}

= - \frac{y x}{( y ^{2} - x ^{2} ) ^{\frac{3}{2}}}

\int 2 x - 3 d x

f (x) = - x + \frac{7}{x} + 1 at x

(x^{2} \cdot 5^{x})^{7}

x \to 2 lim (\frac{x ^{2} + 4}{x + 2})

(4 x^{2} + 5) (4 x - 3)

What is (\partial)/(\partial x)(-y/(sqrt(y^2-x^2))) ?
The answer to (\partial)/(\partial x)(-y/(sqrt(y^2-x^2))) is -(yx)/((y^2-x^2)^{3/2)}