What is (\partial)/(\partial x)((42)/((5+x^2+y^2))) ?

The answer to (\partial)/(\partial x)((42)/((5+x^2+y^2))) is -(84x)/((5+x^2+y^2)^2)

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(\partial)/(\partial x)((42)/((5+x^2+y^2)))

\frac{\partial}{\partial x} (\frac{42}{( 5 + x ^{2} + y ^{2} )})

- \frac{84 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{42}{( 5 + x ^{2} + y ^{2} )})

Treat

y

as a constant

Take the constant out:

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

= 42 \frac{\partial}{\partial x} (\frac{1}{5 + x ^{2} + y ^{2}})

Apply exponent rule:

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

= 42 \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

= 42 (- \frac{1}{( 5 + x ^{2} + y ^{2} ) ^{2}} \cdot 2 x)

Simplify

42 (- \frac{1}{( 5 + x ^{2} + y ^{2} ) ^{2}} \cdot 2 x) : - \frac{84 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

= - \frac{84 x}{( 5 + x ^{2} + y ^{2} ) ^{2}}

\int \frac{1}{1 + cos ( 2 x )} d x

\int x (x^{2} + 1)^{5} d x

x = 3 y^{2}, x = 4 + 2 y^{2}

x \to - 4 lim (x^{2} - 3)

x^{2} y^{^{''}} + 3 x y^{^{'}} - 24 y = 0

What is (\partial)/(\partial x)((42)/((5+x^2+y^2))) ?
The answer to (\partial)/(\partial x)((42)/((5+x^2+y^2))) is -(84x)/((5+x^2+y^2)^2)