Calculus Examples

$g (t) = 4 t^{- \frac{3}{8}}$

Step 1

Since $4$ is constant with respect to $t$ , the derivative of $4 t^{- \frac{3}{8}}$ with respect to $t$ is $4 \frac{d}{d t} [t^{- \frac{3}{8}}]$ .

$4 \frac{d}{d t} [t^{- \frac{3}{8}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - \frac{3}{8}$ .

$4 (- \frac{3}{8} t^{- \frac{3}{8} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$4 (- \frac{3}{8} t^{- \frac{3}{8} - 1 \cdot \frac{8}{8}})$

Step 4

Combine $- 1$ and $\frac{8}{8}$ .

$4 (- \frac{3}{8} t^{- \frac{3}{8} + \frac{- 1 \cdot 8}{8}})$

Step 5

Combine the numerators over the common denominator.

$4 (- \frac{3}{8} t^{\frac{- 3 - 1 \cdot 8}{8}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $8$ .

$4 (- \frac{3}{8} t^{\frac{- 3 - 8}{8}})$

Step 6.2

Subtract $8$ from $- 3$ .

$4 (- \frac{3}{8} t^{\frac{- 11}{8}})$

Step 7

Move the negative in front of the fraction.

$4 (- \frac{3}{8} t^{- \frac{11}{8}})$

Step 8

Combine $t^{- \frac{11}{8}}$ and $\frac{3}{8}$ .

$4 (- \frac{t^{- \frac{11}{8}} \cdot 3}{8})$

Step 9

Multiply $- 1$ by $4$ .

$- 4 \frac{t^{- \frac{11}{8}} \cdot 3}{8}$

Step 10

Combine $- 4$ and $\frac{t^{- \frac{11}{8}} \cdot 3}{8}$ .

$\frac{- 4 (t^{- \frac{11}{8}} \cdot 3)}{8}$

Step 11

Multiply.

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Step 11.1

Multiply $3$ by $- 4$ .

$\frac{- 12 t^{- \frac{11}{8}}}{8}$

Step 11.2

Move $t^{- \frac{11}{8}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 12}{8 t^{\frac{11}{8}}}$

Step 12

Factor $4$ out of $- 12$ .

$\frac{4 (- 3)}{8 t^{\frac{11}{8}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $4$ out of $8 t^{\frac{11}{8}}$ .

$\frac{4 (- 3)}{4 (2 t^{\frac{11}{8}})}$

Step 13.2

Cancel the common factor.

$\frac{4 \cdot - 3}{4 (2 t^{\frac{11}{8}})}$

Step 13.3

Rewrite the expression.

$\frac{- 3}{2 t^{\frac{11}{8}}}$

Step 14

Move the negative in front of the fraction.

$- \frac{3}{2 t^{\frac{11}{8}}}$