Calculus Examples

$t + \frac{1}{t}$

Step 1

Differentiate.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $t + \frac{1}{t}$ with respect to $t$ is $\frac{d}{d t} [t] + \frac{d}{d t} [\frac{1}{t}]$ .

$\frac{d}{d t} [t] + \frac{d}{d t} [\frac{1}{t}]$

Step 1.2

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

$1 + \frac{d}{d t} [\frac{1}{t}]$

Step 2

Evaluate $\frac{d}{d t} [\frac{1}{t}]$ .

Tap for more steps...

Step 2.1

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

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

Step 2.2

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

$1 - t^{- 2}$

Step 3

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

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

Step 4

Reorder terms.

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