Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Cancel the common factor.

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

Step 12

Rewrite the expression.

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