What is (\partial)/(\partial x)((7x-5y)/(y^2+x^2)) ?

The answer to (\partial)/(\partial x)((7x-5y)/(y^2+x^2)) is (7y^2-7x^2+10xy)/((y^2+x^2)^2)

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(\partial)/(\partial x)((7x-5y)/(y^2+x^2))

\frac{\partial}{\partial x} (\frac{7 x - 5 y}{y ^{2} + x ^{2}})

\frac{7 y ^{2} - 7 x ^{2} + 10 x y}{( y ^{2} + x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{7 x - 5 y}{y ^{2} + x ^{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} ( 7 x - 5 y ) ( y ^{2} + x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + x ^{2} ) ( 7 x - 5 y )}{( y ^{2} + x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (7 x - 5 y) = 7

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

= \frac{7 ( y ^{2} + x ^{2} ) - 2 x ( 7 x - 5 y )}{( y ^{2} + x ^{2} ) ^{2}}

Expand

7 (y^{2} + x^{2}) - 2 x (7 x - 5 y) : 7 y^{2} - 7 x^{2} + 10 x y

= \frac{7 y ^{2} - 7 x ^{2} + 10 x y}{( y ^{2} + x ^{2} ) ^{2}}

x \to - π lim ((x + π) csc (x))

8 t \frac{d y}{d t} + y = t^{7}

\frac{d}{d x} (\frac{x ^{2}}{x ^{4} + x ^{2} + 1})

\frac{d}{d x} (arcsin (sin (x) - cos (x)))

\frac{10 x ^{3}}{3} + \frac{13 x ^{2}}{2} + 4 x + 16

What is (\partial)/(\partial x)((7x-5y)/(y^2+x^2)) ?
The answer to (\partial)/(\partial x)((7x-5y)/(y^2+x^2)) is (7y^2-7x^2+10xy)/((y^2+x^2)^2)