What is (\partial)/(\partial x)((((x+4))/(y+5))^{1/2}) ?

The answer to (\partial)/(\partial x)((((x+4))/(y+5))^{1/2}) is 1/(2(x+4)^{1/2)(y+5)^{1/2}}

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(\partial)/(\partial x)((((x+4))/(y+5))^{1/2})

\frac{\partial}{\partial x} ((\frac{( x + 4 )}{y + 5})^{\frac{1}{2}})

\frac{1}{2 ( x + 4 ) ^{\frac{1}{2}} ( y + 5 ) ^{\frac{1}{2}}}

Solution steps

\frac{\partial}{\partial x} ((\frac{( x + 4 )}{y + 5})^{\frac{1}{2}})

Treat

y

as a constant

Apply the chain rule:

\frac{1}{2 ( \frac{x + 4}{y + 5} ) ^{\frac{1}{2}}} \frac{\partial}{\partial x} (\frac{x + 4}{y + 5})

= \frac{1}{2 ( \frac{x + 4}{y + 5} ) ^{\frac{1}{2}}} \frac{\partial}{\partial x} (\frac{x + 4}{y + 5})

\frac{\partial}{\partial x} (\frac{x + 4}{y + 5}) = \frac{1}{y + 5}

= \frac{1}{2 ( \frac{x + 4}{y + 5} ) ^{\frac{1}{2}}} \cdot \frac{1}{y + 5}

Simplify

\frac{1}{2 ( \frac{x + 4}{y + 5} ) ^{\frac{1}{2}}} \cdot \frac{1}{y + 5} : \frac{1}{2 ( x + 4 ) ^{\frac{1}{2}} ( y + 5 ) ^{\frac{1}{2}}}

= \frac{1}{2 ( x + 4 ) ^{\frac{1}{2}} ( y + 5 ) ^{\frac{1}{2}}}

What is (\partial)/(\partial x)((((x+4))/(y+5))^{1/2}) ?
The answer to (\partial)/(\partial x)((((x+4))/(y+5))^{1/2}) is 1/(2(x+4)^{1/2)(y+5)^{1/2}}