Calculus Examples

$f (t) = t^{- 5}$ ;, $(\frac{1}{2}, 32)$

Step 1

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = - 5$ .

$- 5 t^{- 6}$

Step 2

Simplify.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \frac{1}{t^{6}}$

Step 2.2

Combine $- 5$ and $\frac{1}{t^{6}}$ .

$\frac{- 5}{t^{6}}$

Step 2.3

Move the negative in front of the fraction.

$- \frac{5}{t^{6}}$

Step 3

Evaluate the derivative at $t = \frac{1}{2}$ .

$- \frac{5}{{(\frac{1}{2})}^{6}}$

Step 4

Simplify.

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Step 4.1

Simplify the denominator.

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Step 4.1.1

Apply the product rule to $\frac{1}{2}$ .

$- \frac{5}{\frac{1^{6}}{2^{6}}}$

Step 4.1.2

One to any power is one.

$- \frac{5}{\frac{1}{2^{6}}}$

Step 4.1.3

Raise $2$ to the power of $6$ .

$- \frac{5}{\frac{1}{64}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- (5 \cdot 64)$

Step 4.3

Multiply $- (5 \cdot 64)$ .

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Step 4.3.1

Multiply $5$ by $64$ .

$- 1 \cdot 320$

Step 4.3.2

Multiply $- 1$ by $320$ .

$- 320$