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

The answer to (\partial)/(\partial x)((-x)/((x+y)^2)) is -(-x+y)/((x+y)^3)

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

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

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

Solution steps

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

Simplify

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

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

- \frac{1 \cdot ( x + y ) ^{2} - 2 ( x + y ) x}{( ( x + y ) ^{2} ) ^{2}} : - \frac{- x + y}{( x + y ) ^{3}}

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

f (x) = sech (x)

(\frac{x ^{2} + 5}{x ^{2} - 5})^{2}

\int sin (0)

\int (6 x^{2} - 8 x + 3) d x

\frac{d}{d x} (2 x tan (x) + x^{2} sec^{2} (x))

What is (\partial)/(\partial x)((-x)/((x+y)^2)) ?
The answer to (\partial)/(\partial x)((-x)/((x+y)^2)) is -(-x+y)/((x+y)^3)