What is (\partial)/(\partial x)(z/(x+4y)) ?

The answer to (\partial)/(\partial x)(z/(x+4y)) is -z/((x+4y)^2)

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(\partial)/(\partial x)(z/(x+4y))

\frac{\partial}{\partial x} (\frac{z}{x + 4 y})

- \frac{z}{( x + 4 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{z}{x + 4 y})

Treat

z, y

as constants

Take the constant out:

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

= z \frac{\partial}{\partial x} (\frac{1}{x + 4 y})

Apply exponent rule:

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

= z \frac{\partial}{\partial x} ((x + 4 y)^{- 1})

Apply the chain rule:

- \frac{1}{( x + 4 y ) ^{2}} \frac{\partial}{\partial x} (x + 4 y)

= - \frac{1}{( x + 4 y ) ^{2}} \frac{\partial}{\partial x} (x + 4 y)

\frac{\partial}{\partial x} (x + 4 y) = 1

= z (- \frac{1}{( x + 4 y ) ^{2}} \cdot 1)

Simplify

z (- \frac{1}{( x + 4 y ) ^{2}} \cdot 1) : - \frac{z}{( x + 4 y ) ^{2}}

= - \frac{z}{( x + 4 y ) ^{2}}

What is (\partial)/(\partial x)(z/(x+4y)) ?
The answer to (\partial)/(\partial x)(z/(x+4y)) is -z/((x+4y)^2)