Calculus Examples

$\ln^{2} (t)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = \ln (t)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (t)$ .

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

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 1.3

Replace all occurrences of $u$ with $\ln (t)$ .

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

Step 2

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

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

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{1}{t}$ and $2$ .

$\frac{2}{t} \ln (t)$

Step 3.2

Combine $\frac{2}{t}$ and $\ln (t)$ .

$\frac{2 \ln (t)}{t}$

Step 4

Simplify $2 \ln (t)$ by moving $2$ inside the logarithm.

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