Calculus Examples

$f (t) = 2 t^{- \frac{1}{4}}$

Step 1

Since $2$ is constant with respect to $t$ , the derivative of $2 t^{- \frac{1}{4}}$ with respect to $t$ is $2 \frac{d}{d t} [t^{- \frac{1}{4}}]$ .

$2 \frac{d}{d t} [t^{- \frac{1}{4}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - \frac{1}{4}$ .

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

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$2 (- \frac{1}{4} t^{- \frac{1}{4} - 1 \cdot \frac{4}{4}})$

Step 4

Combine $- 1$ and $\frac{4}{4}$ .

$2 (- \frac{1}{4} t^{- \frac{1}{4} + \frac{- 1 \cdot 4}{4}})$

Step 5

Combine the numerators over the common denominator.

$2 (- \frac{1}{4} t^{\frac{- 1 - 1 \cdot 4}{4}})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $4$ .

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

Step 6.2

Subtract $4$ from $- 1$ .

$2 (- \frac{1}{4} t^{\frac{- 5}{4}})$

Step 7

Move the negative in front of the fraction.

$2 (- \frac{1}{4} t^{- \frac{5}{4}})$

Step 8

Combine $t^{- \frac{5}{4}}$ and $\frac{1}{4}$ .

$2 (- \frac{t^{- \frac{5}{4}}}{4})$

Step 9

Multiply $- 1$ by $2$ .

$- 2 \frac{t^{- \frac{5}{4}}}{4}$

Step 10

Combine $- 2$ and $\frac{t^{- \frac{5}{4}}}{4}$ .

$\frac{- 2 t^{- \frac{5}{4}}}{4}$

Step 11

Move $t^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 2}{4 t^{\frac{5}{4}}}$

Step 12

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{4 t^{\frac{5}{4}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $4 t^{\frac{5}{4}}$ .

$\frac{2 (- 1)}{2 (2 t^{\frac{5}{4}})}$

Step 13.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 (2 t^{\frac{5}{4}})}$

Step 13.3

Rewrite the expression.

$\frac{- 1}{2 t^{\frac{5}{4}}}$

Step 14

Move the negative in front of the fraction.

$- \frac{1}{2 t^{\frac{5}{4}}}$