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

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

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

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

\frac{1}{2 x + 4 y + 4}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

= \frac{1}{2 \frac{x + 4}{y + 4}} \cdot \frac{1}{y + 4}

Simplify

\frac{1}{2 \frac{x + 4}{y + 4}} \cdot \frac{1}{y + 4} : \frac{1}{2 x + 4 y + 4}

= \frac{1}{2 x + 4 y + 4}

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