Calculus Examples

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

Step 1

Since $\frac{2}{3}$ is constant with respect to $t$ , the derivative of $\frac{2 t^{\frac{9}{2}}}{3}$ with respect to $t$ is $\frac{2}{3} \frac{d}{d t} [t^{\frac{9}{2}}]$ .

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{9}{2}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $9$ .

$\frac{2}{3} (\frac{9}{2} t^{\frac{7}{2}})$

Step 7

Combine $\frac{9}{2}$ and $t^{\frac{7}{2}}$ .

$\frac{2}{3} \cdot \frac{9 t^{\frac{7}{2}}}{2}$

Step 8

Multiply $\frac{2}{3}$ by $\frac{9 t^{\frac{7}{2}}}{2}$ .

$\frac{2 (9 t^{\frac{7}{2}})}{3 \cdot 2}$

Step 9

Multiply.

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Step 9.1

Multiply $9$ by $2$ .

$\frac{18 t^{\frac{7}{2}}}{3 \cdot 2}$

Step 9.2

Multiply $3$ by $2$ .

$\frac{18 t^{\frac{7}{2}}}{6}$

Step 10

Factor $6$ out of $18 t^{\frac{7}{2}}$ .

$\frac{6 (3 t^{\frac{7}{2}})}{6}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $6$ out of $6$ .

$\frac{6 (3 t^{\frac{7}{2}})}{6 (1)}$

Step 11.2

Cancel the common factor.

$\frac{6 (3 t^{\frac{7}{2}})}{6 \cdot 1}$

Step 11.3

Rewrite the expression.

$\frac{3 t^{\frac{7}{2}}}{1}$

Step 11.4

Divide $3 t^{\frac{7}{2}}$ by $1$ .

$3 t^{\frac{7}{2}}$