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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

\frac{1}{\frac{x ^{2} + y ^{2}}{x + y}} \frac{\partial}{\partial x} (\frac{x ^{2} + y ^{2}}{x + y})

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

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

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

Simplify

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

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

x \to 1 lim (\frac{ln ( 1 - x ^{2} )}{x})

x \to 1 lim (4)

y = x^{2} + 3 x + 5

\frac{\partial}{\partial x} (arctan (9 x - 9 y))

\int \frac{9 x ^{2} - 4}{x} d x

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