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

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

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(\partial)/(\partial x)((xy)/(x^2+z^2))

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{- 8 x + 2}{3 x + 5})

x \to - 13 lim (- 12)

f (x) = \frac{x ^{3} + 2}{3}

\frac{4}{x}, at x = 3

\frac{s ^{2} + 1}{s + 5}

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