Calculus Examples

$g (t) = \frac{9}{\sqrt{t}}$

Step 1

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

$\frac{d}{d t} [\frac{9}{t^{\frac{1}{2}}}]$

Step 2

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

$9 \frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}}]$

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

$9 (- \frac{1}{2} t^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 6

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

$9 (- \frac{1}{2} t^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$9 (- \frac{1}{2} t^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

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

Step 8.2

Subtract $2$ from $- 1$ .

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

Multiply $- 1$ by $9$ .

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

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

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

Step 13.2

Move the negative in front of the fraction.

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