Calculus Examples

$\frac{\sqrt{t}}{2}$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the expression.

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Step 11.1

Multiply $2$ by $2$ .

$\frac{t^{- \frac{1}{2}}}{4}$

Step 11.2

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

$\frac{1}{4 t^{\frac{1}{2}}}$