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

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

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

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

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

Solution steps

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

Treat

x, z

as constants

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

n = 1 \sum \infty (- 0.75)^{n}

\int x (x - 5) d x

y = 2 x^{2} ln (x), y = 8 ln (x)

\int \frac{48}{( t + 1 ) ^{4}} - 3 d t

\frac{d y}{d x} = \frac{x ^{7} - 8 y}{x}

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