Calculus Examples

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

Step 1

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

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

Step 2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

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

Step 3

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 $g (t) = \ln (t)$ .

Tap for more steps...

Step 3.1

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

$\ln^{2} (t) (- \sin (t)) + \cos (t) (\frac{d}{d u} [u^{2}] \frac{d}{d t} [\ln (t)])$

Step 3.2

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

$\ln^{2} (t) (- \sin (t)) + \cos (t) (2 u \frac{d}{d t} [\ln (t)])$

Step 3.3

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

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

Step 4

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

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

Step 5

Combine fractions.

Tap for more steps...

Step 5.1

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

$\ln^{2} (t) (- \sin (t)) + \cos (t) (\frac{2}{t} \ln (t))$

Step 5.2

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

$\ln^{2} (t) (- \sin (t)) + \cos (t) \frac{2 \ln (t)}{t}$

Step 5.3

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

$\ln^{2} (t) (- \sin (t)) + \frac{\cos (t) (2 \ln (t))}{t}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Reorder terms.

$- \ln^{2} (t) \sin (t) + \frac{\cos (t) (2 \ln (t))}{t}$

Step 6.2

Move $2$ to the left of $\cos (t)$ .

$- \ln^{2} (t) \sin (t) + \frac{2 \cos (t) \ln (t)}{t}$