Calculus Examples

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

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (t^{2})$ .

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

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d t} [\sin (t^{2})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (t^{2})$ .

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

Step 2

Convert from $\frac{1}{\sin (t^{2})}$ to $\csc (t^{2})$ .

$\csc (t^{2}) \frac{d}{d t} [\sin (t^{2})]$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $t^{2}$ .

$\csc (t^{2}) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d t} [t^{2}])$

Step 3.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$\csc (t^{2}) (\cos (u_{2}) \frac{d}{d t} [t^{2}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $t^{2}$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\csc (t^{2}) \cos (t^{2}) (2 t)$ .

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

Step 5.2

Rewrite $\csc (t^{2})$ in terms of sines and cosines.

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

Step 5.3

Multiply $2 t \cos (t^{2}) \frac{1}{\sin (t^{2})}$ .

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Step 5.3.1

Combine $2$ and $\frac{1}{\sin (t^{2})}$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

Separate fractions.

$\frac{2 t}{1} \cdot \frac{\cos (t^{2})}{\sin (t^{2})}$

Step 5.5

Convert from $\frac{\cos (t^{2})}{\sin (t^{2})}$ to $\cot (t^{2})$ .

$\frac{2 t}{1} \cot (t^{2})$

Step 5.6

Divide $2 t$ by $1$ .

$2 t \cot (t^{2})$