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

The answer to (\partial)/(\partial x)((2x^2)/(y^2+x^2)) is (4xy^2)/((y^2+x^2)^2)

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

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

\frac{4 x y ^{2}}{( y ^{2} + x ^{2} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

2 \cdot \frac{2 x ( y ^{2} + x ^{2} ) - 2 x x ^{2}}{( y ^{2} + x ^{2} ) ^{2}} : \frac{4 x y ^{2}}{( y ^{2} + x ^{2} ) ^{2}}

= \frac{4 x y ^{2}}{( y ^{2} + x ^{2} ) ^{2}}

\int_{- 1}^{3} 2 x + 3 - x^{2} d x

\frac{d}{d x} ((arctan (6 x))^{2})

x \to 0 lim (\frac{6 x ^{2} + 6 x}{x})

f (x) = 4 x^{- \frac{1}{8}}

What is (\partial)/(\partial x)((2x^2)/(y^2+x^2)) ?
The answer to (\partial)/(\partial x)((2x^2)/(y^2+x^2)) is (4xy^2)/((y^2+x^2)^2)