What is (\partial)/(\partial x)((82)/(5+x^2+y^2)) ?

The answer to (\partial)/(\partial x)((82)/(5+x^2+y^2)) is -(164x)/((5+x^2+y^2)^2)

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(\partial)/(\partial x)((82)/(5+x^2+y^2))

\frac{\partial}{\partial x} (\frac{82}{5 + x ^{2} + y ^{2}})

- \frac{164 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{82}{5 + x ^{2} + y ^{2}})

Treat

y

as a constant

Take the constant out:

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

= 82 \frac{\partial}{\partial x} (\frac{1}{5 + x ^{2} + y ^{2}})

Apply exponent rule:

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

= 82 \frac{\partial}{\partial x} ((5 + x^{2} + y^{2})^{- 1})

Apply the chain rule:

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

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

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

= 82 (- \frac{1}{( 5 + x ^{2} + y ^{2} ) ^{2}} \cdot 2 x)

Simplify

82 (- \frac{1}{( 5 + x ^{2} + y ^{2} ) ^{2}} \cdot 2 x) : - \frac{164 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

= - \frac{164 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

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What is (\partial)/(\partial x)((82)/(5+x^2+y^2)) ?
The answer to (\partial)/(\partial x)((82)/(5+x^2+y^2)) is -(164x)/((5+x^2+y^2)^2)