What is the d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) ?

The d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) is -({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}

What is the first d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) ?

The first d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) is -({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}

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

- \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + y^{2} + z^{2})

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

= - \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x

Simplify

- \frac{1}{2 ( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}} \cdot 2 x : - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

= - \frac{x}{( x ^{2} + y ^{2} + z ^{2} ) ^{\frac{3}{2}}}

What is the d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) ?
The d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) is -({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}
What is the first d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) ?
The first d/(d{x)}(1/(sqrt({x)^2+{y)^2+{z}^2}}) is -({x})/(({x)^2+{y}^2+{z}^2)^{3/2}}