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

The answer to (\partial)/(\partial x)(-(cx)/(x^2+y^2)) is -(c(-x^2+y^2))/((x^2+y^2)^2)

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

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

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

Solution steps

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

Treat

c, y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

\int_{1}^{3} \frac{1}{4 + ( x - 1 ) ^{2}} d x

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\int 24 x y^{2} + 3 y e^{x y} d x

\frac{d y}{d x} = e^{(6 x - 3 y)}, y (0) = 3

\int (2 - e^{x})^{2} d x

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