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

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

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(\partial)/(\partial x)((x^2y^2)/(x+y))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (30 x^{4} y^{5} + 24 x^{5} y^{7})

f (x) = \frac{1}{( x - 2 )}

cos (a^{9} + x^{9})

x \to 0 lim (cos (\frac{1}{x}))

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