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

The answer to (\partial)/(\partial x)((yz)/(x^2+y^2)) is -(2yzx)/((x^2+y^2)^2)

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(\partial)/(\partial x)((yz)/(x^2+y^2))

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d x} (\frac{x ^{4}}{( 1 + 2 x ) ^{3}})

\frac{x ^{6} - 1}{x - 1}

\int \frac{1}{2 x ^{2} + 4 x + 5} d x

\frac{d}{d x} (\frac{x ^{2} + 6}{sin ^{4} ( x )})

f (x) = - 9 e^{x^{- 9}}

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