What is (\partial)/(\partial x)(1/((x^2+y^2+z^2))) ?

The answer to (\partial)/(\partial x)(1/((x^2+y^2+z^2))) is -(2x)/((x^2+y^2+z^2)^2)

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(\partial)/(\partial x)(1/((x^2+y^2+z^2)))

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

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

Solution steps

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

Treat

y, z

as constants

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int \frac{7 sin ( 2 x )}{sin ( x )} d x

\frac{d y}{d t} = \frac{3}{t} y + t^{5}

x \to 8 lim (- 6 x)

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

\int \frac{- 2 x - 1}{( x ^{2} + x + 1 ) ^{6}} d x

What is (\partial)/(\partial x)(1/((x^2+y^2+z^2))) ?
The answer to (\partial)/(\partial x)(1/((x^2+y^2+z^2))) is -(2x)/((x^2+y^2+z^2)^2)