Calculus Examples

${(1 - {(\frac{v}{c})}^{2})}^{- \frac{1}{2}}$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $1 - {(\frac{v}{c})}^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{- \frac{1}{2}}] \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 1.2

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

$- \frac{1}{2} {u_{1}}^{- \frac{1}{2} - 1} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $1 - {(\frac{v}{c})}^{2}$ .

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{- \frac{1}{2} - 1} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 2

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

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 3

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

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 4

Combine the numerators over the common denominator.

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{- 1 - 2}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 5.2

Subtract $2$ from $- 1$ .

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{- 3}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 6

Differentiate.

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Step 6.1

Move the negative in front of the fraction.

$- \frac{1}{2} {(1 - {(\frac{v}{c})}^{2})}^{- \frac{3}{2}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 6.2

Combine fractions.

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Step 6.2.1

Combine ${(1 - {(\frac{v}{c})}^{2})}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$- \frac{{(1 - {(\frac{v}{c})}^{2})}^{- \frac{3}{2}}}{2} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 6.2.2

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

$- \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [1 - {(\frac{v}{c})}^{2}]$

Step 6.3

By the Sum Rule, the derivative of $1 - {(\frac{v}{c})}^{2}$ with respect to $v$ is $\frac{d}{d v} [1] + \frac{d}{d v} [- {(\frac{v}{c})}^{2}]$ .

$- \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (\frac{d}{d v} [1] + \frac{d}{d v} [- {(\frac{v}{c})}^{2}])$

Step 6.4

Since $1$ is constant with respect to $v$ , the derivative of $1$ with respect to $v$ is $0$ .

$- \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (0 + \frac{d}{d v} [- {(\frac{v}{c})}^{2}])$

Step 6.5

Add $0$ and $\frac{d}{d v} [- {(\frac{v}{c})}^{2}]$ .

$- \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [- {(\frac{v}{c})}^{2}]$

Step 6.6

Since $- 1$ is constant with respect to $v$ , the derivative of $- {(\frac{v}{c})}^{2}$ with respect to $v$ is $- \frac{d}{d v} [{(\frac{v}{c})}^{2}]$ .

$- \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (- \frac{d}{d v} [{(\frac{v}{c})}^{2}])$

Step 6.7

Multiply.

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Step 6.7.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [{(\frac{v}{c})}^{2}]$

Step 6.7.2

Multiply $\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}}$ by $1$ .

$\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [{(\frac{v}{c})}^{2}]$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d v} [f (g (v))]$ is $f' (g (v)) g' (v)$ where $f (v) = v^{2}$ and $g (v) = \frac{v}{c}$ .

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Step 7.1

To apply the Chain Rule, set $u_{2}$ as $\frac{v}{c}$ .

$\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d v} [\frac{v}{c}])$

Step 7.2

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

$\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (2 u_{2} \frac{d}{d v} [\frac{v}{c}])$

Step 7.3

Replace all occurrences of $u_{2}$ with $\frac{v}{c}$ .

$\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (2 \frac{v}{c} \frac{d}{d v} [\frac{v}{c}])$

Step 8

Differentiate using the Constant Multiple Rule.

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Step 8.1

Combine $2$ and $\frac{v}{c}$ .

$\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} (\frac{2 v}{c} \frac{d}{d v} [\frac{v}{c}])$

Step 8.2

Simplify terms.

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Step 8.2.1

Multiply $\frac{2 v}{c}$ by $\frac{1}{2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}}$ .

$\frac{2 v}{c (2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}})} \frac{d}{d v} [\frac{v}{c}]$

Step 8.2.2

Cancel the common factor.

$\frac{2 v}{c (2 {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}})} \frac{d}{d v} [\frac{v}{c}]$

Step 8.2.3

Rewrite the expression.

$\frac{v}{c ({(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}})} \frac{d}{d v} [\frac{v}{c}]$

Step 8.3

Since $\frac{1}{c}$ is constant with respect to $v$ , the derivative of $\frac{v}{c}$ with respect to $v$ is $\frac{1}{c} \frac{d}{d v} [v]$ .

$\frac{v}{c ({(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}})} (\frac{1}{c} \frac{d}{d v} [v])$

Step 8.4

Multiply $\frac{1}{c}$ by $\frac{v}{c ({(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}})}$ .

$\frac{v}{c (c ({(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}))} \frac{d}{d v} [v]$

Step 9

Raise $c$ to the power of $1$ .

$\frac{v}{c^{1} c {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [v]$

Step 10

Raise $c$ to the power of $1$ .

$\frac{v}{c^{1} c^{1} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [v]$

Step 11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{v}{c^{1 + 1} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [v]$

Step 12

Add $1$ and $1$ .

$\frac{v}{c^{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \frac{d}{d v} [v]$

Step 13

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

$\frac{v}{c^{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}} \cdot 1$

Step 14

Simplify the expression.

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Step 14.1

Multiply $\frac{v}{c^{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}}$ by $1$ .

$\frac{v}{c^{2} {(1 - {(\frac{v}{c})}^{2})}^{\frac{3}{2}}}$

Step 14.2

Apply the product rule to $\frac{v}{c}$ .

$\frac{v}{c^{2} {(1 - \frac{v^{2}}{c^{2}})}^{\frac{3}{2}}}$