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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to 5 lim (\frac{x - 5}{x ^{2} - 5})

\int \frac{tan ^{4} ( x )}{cos ^{2} ( x )} d x

\frac{d y}{d x} = y e^{x}, y (0) = 2 e

\frac{d}{d x} (sin (y x))

\frac{d}{d t} (1 + t^{2})

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