What is (\partial)/(\partial x)((6y)/(x^2+y^2+5)) ?

The answer to (\partial)/(\partial x)((6y)/(x^2+y^2+5)) is -(12yx)/((x^2+y^2+5)^2)

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(\partial)/(\partial x)((6y)/(x^2+y^2+5))

\frac{\partial}{\partial x} (\frac{6 y}{x ^{2} + y ^{2} + 5})

- \frac{12 y x}{( x ^{2} + y ^{2} + 5 ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{6 y}{x ^{2} + y ^{2} + 5})

Treat

y

as a constant

Take the constant out:

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

= 6 y \frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2} + 5})

Apply exponent rule:

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

= 6 y \frac{\partial}{\partial x} ((x^{2} + y^{2} + 5)^{- 1})

Apply the chain rule:

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

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

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

= 6 y (- \frac{1}{( x ^{2} + y ^{2} + 5 ) ^{2}} \cdot 2 x)

Simplify

6 y (- \frac{1}{( x ^{2} + y ^{2} + 5 ) ^{2}} \cdot 2 x) : - \frac{12 y x}{( x ^{2} + y ^{2} + 5 ) ^{2}}

= - \frac{12 y x}{( x ^{2} + y ^{2} + 5 ) ^{2}}

What is (\partial)/(\partial x)((6y)/(x^2+y^2+5)) ?
The answer to (\partial)/(\partial x)((6y)/(x^2+y^2+5)) is -(12yx)/((x^2+y^2+5)^2)