What is the derivative of (x^2/(x^2+y^2)) ?

The derivative of (x^2/(x^2+y^2)) is (-2x^2y\frac{dy)/(dx)+2xy^2}{(x^2+y^2)^2}

What is the first derivative of (x^2/(x^2+y^2)) ?

The first derivative of (x^2/(x^2+y^2)) is (-2x^2y\frac{dy)/(dx)+2xy^2}{(x^2+y^2)^2}

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

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

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

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

Expand

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

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

\frac{d}{d x} (sin^{\frac{1}{2}} (x))

\frac{d}{d x} (10 x + 9 x^{2} + 3 y)

\frac{d y}{d x}, y = arcsin (2 x + 1)

e^{\frac{t ^{2}}{2}}

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

What is the derivative of (x^2/(x^2+y^2)) ?
The derivative of (x^2/(x^2+y^2)) is (-2x^2y\frac{dy)/(dx)+2xy^2}{(x^2+y^2)^2}
What is the first derivative of (x^2/(x^2+y^2)) ?
The first derivative of (x^2/(x^2+y^2)) is (-2x^2y\frac{dy)/(dx)+2xy^2}{(x^2+y^2)^2}