Calculus Examples

$f (t) = \frac{6}{\sqrt{t^{2} - 9}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t^{2} - 9}$ as ${(t^{2} - 9)}^{\frac{1}{2}}$ .

$\frac{d}{d t} [\frac{6}{{(t^{2} - 9)}^{\frac{1}{2}}}]$

Step 1.2

Since $6$ is constant with respect to $t$ , the derivative of $\frac{6}{{(t^{2} - 9)}^{\frac{1}{2}}}$ with respect to $t$ is $6 \frac{d}{d t} [\frac{1}{{(t^{2} - 9)}^{\frac{1}{2}}}]$ .

$6 \frac{d}{d t} [\frac{1}{{(t^{2} - 9)}^{\frac{1}{2}}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Rewrite $\frac{1}{{(t^{2} - 9)}^{\frac{1}{2}}}$ as ${({(t^{2} - 9)}^{\frac{1}{2}})}^{- 1}$ .

$6 \frac{d}{d t} [{({(t^{2} - 9)}^{\frac{1}{2}})}^{- 1}]$

Step 1.3.2

Multiply the exponents in ${({(t^{2} - 9)}^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 \frac{d}{d t} [{(t^{2} - 9)}^{\frac{1}{2} \cdot - 1}]$

Step 1.3.2.2

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

$6 \frac{d}{d t} [{(t^{2} - 9)}^{\frac{- 1}{2}}]$

Step 1.3.2.3

Move the negative in front of the fraction.

$6 \frac{d}{d t} [{(t^{2} - 9)}^{- \frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- \frac{1}{2}}$ and $g (t) = t^{2} - 9$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $t^{2} - 9$ .

$6 (\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d t} [t^{2} - 9])$

Step 2.2

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

$6 (- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d t} [t^{2} - 9])$

Step 2.3

Replace all occurrences of $u$ with $t^{2} - 9$ .

$6 (- \frac{1}{2} {(t^{2} - 9)}^{- \frac{1}{2} - 1} \frac{d}{d t} [t^{2} - 9])$

Step 3

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

$6 (- \frac{1}{2} {(t^{2} - 9)}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 4

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

$6 (- \frac{1}{2} {(t^{2} - 9)}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 5

Combine the numerators over the common denominator.

$6 (- \frac{1}{2} {(t^{2} - 9)}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$6 (- \frac{1}{2} {(t^{2} - 9)}^{\frac{- 1 - 2}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 6.2

Subtract $2$ from $- 1$ .

$6 (- \frac{1}{2} {(t^{2} - 9)}^{\frac{- 3}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 7

Move the negative in front of the fraction.

$6 (- \frac{1}{2} {(t^{2} - 9)}^{- \frac{3}{2}} \frac{d}{d t} [t^{2} - 9])$

Step 8

Combine ${(t^{2} - 9)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$6 (- \frac{{(t^{2} - 9)}^{- \frac{3}{2}}}{2} \frac{d}{d t} [t^{2} - 9])$

Step 9

Simplify the expression.

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Step 9.1

Move ${(t^{2} - 9)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 (- \frac{1}{2 {(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9])$

Step 9.2

Multiply $- 1$ by $6$ .

$- 6 (\frac{1}{2 {(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9])$

Step 10

Combine $\frac{1}{2 {(t^{2} - 9)}^{\frac{3}{2}}}$ and $- 6$ .

$\frac{- 6}{2 {(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9]$

Step 11

Factor $2$ out of $- 6$ .

$\frac{2 \cdot - 3}{2 {(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9]$

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(t^{2} - 9)}^{\frac{3}{2}}$ .

$\frac{2 \cdot - 3}{2 ({(t^{2} - 9)}^{\frac{3}{2}})} \frac{d}{d t} [t^{2} - 9]$

Step 12.2

Cancel the common factor.

$\frac{2 \cdot - 3}{2 {(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9]$

Step 12.3

Rewrite the expression.

$\frac{- 3}{{(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9]$

Step 13

Move the negative in front of the fraction.

$- \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} \frac{d}{d t} [t^{2} - 9]$

Step 14

By the Sum Rule, the derivative of $t^{2} - 9$ with respect to $t$ is $\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 9]$ .

$- \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} (\frac{d}{d t} [t^{2}] + \frac{d}{d t} [- 9])$

Step 15

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

$- \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} (2 t + \frac{d}{d t} [- 9])$

Step 16

Since $- 9$ is constant with respect to $t$ , the derivative of $- 9$ with respect to $t$ is $0$ .

$- \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} (2 t + 0)$

Step 17

Combine fractions.

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Step 17.1

Add $2 t$ and $0$ .

$- \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} (2 t)$

Step 17.2

Multiply $2$ by $- 1$ .

$- 2 \frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}} t$

Step 17.3

Combine $- 2$ and $\frac{3}{{(t^{2} - 9)}^{\frac{3}{2}}}$ .

$\frac{- 2 \cdot 3}{{(t^{2} - 9)}^{\frac{3}{2}}} t$

Step 17.4

Multiply $- 2$ by $3$ .

$\frac{- 6}{{(t^{2} - 9)}^{\frac{3}{2}}} t$

Step 17.5

Combine $\frac{- 6}{{(t^{2} - 9)}^{\frac{3}{2}}}$ and $t$ .

$\frac{- 6 t}{{(t^{2} - 9)}^{\frac{3}{2}}}$

Step 17.6

Move the negative in front of the fraction.

$- \frac{6 t}{{(t^{2} - 9)}^{\frac{3}{2}}}$