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

The answer to (\partial)/(\partial z)(x/(sqrt(y+z))) is -x/(2(y+z)^{3/2)}

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

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

- \frac{x}{2 ( y + z ) ^{\frac{3}{2}}}

Solution steps

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

Treat

x, y

as constants

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{1}{2 ( y + z ) ^{\frac{3}{2}}} \frac{\partial}{\partial z} (y + z)

= - \frac{1}{2 ( y + z ) ^{\frac{3}{2}}} \frac{\partial}{\partial z} (y + z)

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

= x (- \frac{1}{2 ( y + z ) ^{\frac{3}{2}}} \cdot 1)

Simplify

x (- \frac{1}{2 ( y + z ) ^{\frac{3}{2}}} \cdot 1) : - \frac{x}{2 ( y + z ) ^{\frac{3}{2}}}

= - \frac{x}{2 ( y + z ) ^{\frac{3}{2}}}

x \to 0 - lim (\frac{( x + 4 )}{( 3 )})

9 y^{^{''}} - 30 y^{^{'}} + 25 y = 0

\int \frac{2}{5} x^{\frac{5}{2}} - \frac{54}{5} d x

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

\int e^{- y x} d x

What is (\partial)/(\partial z)(x/(sqrt(y+z))) ?
The answer to (\partial)/(\partial z)(x/(sqrt(y+z))) is -x/(2(y+z)^{3/2)}