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

The derivative of (x^2y/(x^2+y^2)) is (x^4\frac{dy)/(dx)-x^2y^2(dy)/(dx)+2xy^3}{(x^2+y^2)^2}

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

The first derivative of (x^2y/(x^2+y^2)) is (x^4\frac{dy)/(dx)-x^2y^2(dy)/(dx)+2xy^3}{(x^2+y^2)^2}

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Popular >

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

\frac{x ^{4} \frac{d y}{d x} - x ^{2} y ^{2} \frac{d y}{d x} + 2 x y ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

Solution steps

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

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

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

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

Expand

(2 x y + \frac{d y}{d x} x^{2}) (x^{2} + y^{2}) - (2 x + 2 y \frac{d y}{d x}) x^{2} y : x^{4} \frac{d y}{d x} - x^{2} y^{2} \frac{d y}{d x} + 2 x y^{3}

= \frac{x ^{4} \frac{d y}{d x} - x ^{2} y ^{2} \frac{d y}{d x} + 2 x y ^{3}}{( x ^{2} + y ^{2} ) ^{2}}

y = x^{2} - 8

y^{^{''}} - 3 y^{^{'}} + 2 y = 4 x^{2}

\int \frac{x}{2 x} d x

x \to \infty lim (\frac{x}{k})

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

What is the derivative of (x^2y/(x^2+y^2)) ?
The derivative of (x^2y/(x^2+y^2)) is (x^4\frac{dy)/(dx)-x^2y^2(dy)/(dx)+2xy^3}{(x^2+y^2)^2}
What is the first derivative of (x^2y/(x^2+y^2)) ?
The first derivative of (x^2y/(x^2+y^2)) is (x^4\frac{dy)/(dx)-x^2y^2(dy)/(dx)+2xy^3}{(x^2+y^2)^2}