What is (\partial)/(\partial y)(1/(x+2y)) ?

The answer to (\partial)/(\partial y)(1/(x+2y)) is -2/((x+2y)^2)

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(\partial)/(\partial y)(1/(x+2y))

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

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

Solution steps

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

Treat

x

as a constant

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int_{- 2}^{- 1} x f (x^{2} + 1) d x

f (x) = \frac{x}{( x - 1 ) ^{2}}

\frac{d}{d t} (ln (2 t))

\int_{0}^{1} \frac{1}{x ^{3}} d x

\int \frac{9 t - 5}{t + 1} d t

What is (\partial)/(\partial y)(1/(x+2y)) ?
The answer to (\partial)/(\partial y)(1/(x+2y)) is -2/((x+2y)^2)