Calculus Examples

$g (t) = \frac{1}{\sqrt{t}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Move the negative in front of the fraction.

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

Step 2

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

$- \frac{1}{2} t^{- \frac{1}{2} - 1}$

Step 3

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

$- \frac{1}{2} t^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

$- \frac{1}{2} t^{\frac{- 1 - 1 \cdot 2}{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2} t^{\frac{- 1 - 2}{2}}$

Step 6.2

Subtract $2$ from $- 1$ .

$- \frac{1}{2} t^{\frac{- 3}{2}}$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{2} t^{- \frac{3}{2}}$

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2} \cdot \frac{1}{t^{\frac{3}{2}}}$

Step 8.2

Combine terms.

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Step 8.2.1

Multiply $\frac{1}{t^{\frac{3}{2}}}$ by $\frac{1}{2}$ .

$- \frac{1}{t^{\frac{3}{2}} \cdot 2}$

Step 8.2.2

Move $2$ to the left of $t^{\frac{3}{2}}$ .

$- \frac{1}{2 t^{\frac{3}{2}}}$