Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{2}}$ and $g (x) = x^{2} + y^{2} + z^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} + y^{2} + z^{2}$ .

$\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{1}{2}$ .

$- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} + y^{2} + z^{2}$ .

$- \frac{1}{2} {(x^{2} + y^{2} + z^{2})}^{- \frac{1}{2} - 1} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} {(x^{2} + y^{2} + z^{2})}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 3

Combine $- 1$ and $\frac{2}{2}$ .

$- \frac{1}{2} {(x^{2} + y^{2} + z^{2})}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 4

Combine the numerators over the common denominator.

$- \frac{1}{2} {(x^{2} + y^{2} + z^{2})}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2} {(x^{2} + y^{2} + z^{2})}^{\frac{- 1 - 2}{2}} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 5.2

Subtract $2$ from $- 1$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

Combine ${(x^{2} + y^{2} + z^{2})}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$- \frac{{(x^{2} + y^{2} + z^{2})}^{- \frac{3}{2}}}{2} \frac{d}{d x} [x^{2} + y^{2} + z^{2}]$

Step 6.3

Move ${(x^{2} + y^{2} + z^{2})}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7

By the Sum Rule, the derivative of $x^{2} + y^{2} + z^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [z^{2}]$ .

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

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 9

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$- \frac{1}{2 {(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} (2 x + 0 + \frac{d}{d x} [z^{2}])$

Step 10

Add $2 x$ and $0$ .

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

Step 11

Since $z^{2}$ is constant with respect to $x$ , the derivative of $z^{2}$ with respect to $x$ is $0$ .

$- \frac{1}{2 {(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}} (2 x + 0)$

Step 12

Simplify terms.

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Step 12.1

Add $2 x$ and $0$ .

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

Step 12.2

Multiply $2$ by $- 1$ .

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

Step 12.3

Combine $- 2$ and $\frac{1}{2 {(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}$ .

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

Step 12.4

Combine $\frac{- 2}{2 {(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}}$ and $x$ .

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

Step 12.5

Factor $2$ out of $- 2 x$ .

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

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2 {(x^{2} + y^{2} + z^{2})}^{\frac{3}{2}}$ .

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

Step 13.2

Cancel the common factor.

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

Step 13.3

Rewrite the expression.

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

Step 14

Move the negative in front of the fraction.

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