Calculus Examples

$\frac{\ln (t)}{t}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = \ln (t)$ and $g (t) = t$ .

$\frac{t \frac{d}{d t} [\ln (t)] - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 2

The derivative of $\ln (t)$ with respect to $t$ is $\frac{1}{t}$ .

$\frac{t \frac{1}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Combine $t$ and $\frac{1}{t}$ .

$\frac{\frac{t}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 3.2

Cancel the common factor of $t$ .

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Step 3.2.1

Cancel the common factor.

$\frac{\frac{t}{t} - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 3.2.2

Rewrite the expression.

$\frac{1 - \ln (t) \frac{d}{d t} [t]}{t^{2}}$

Step 3.3

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

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

Step 3.4

Multiply $- 1$ by $1$ .

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