Calculus Examples

$g (t) = \frac{t^{2} \cot (t)}{5}$

Step 1

Since $\frac{1}{5}$ is constant with respect to $t$ , the derivative of $\frac{t^{2} \cot (t)}{5}$ with respect to $t$ is $\frac{1}{5} \frac{d}{d t} [t^{2} \cot (t)]$ .

$\frac{1}{5} \frac{d}{d t} [t^{2} \cot (t)]$

Step 2

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 $f (t) = t^{2}$ and $g (t) = \cot (t)$ .

$\frac{1}{5} (t^{2} \frac{d}{d t} [\cot (t)] + \cot (t) \frac{d}{d t} [t^{2}])$

Step 3

The derivative of $\cot (t)$ with respect to $t$ is $- \csc^{2} (t)$ .

$\frac{1}{5} (t^{2} (- \csc^{2} (t)) + \cot (t) \frac{d}{d t} [t^{2}])$

Step 4

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

$\frac{1}{5} (t^{2} (- \csc^{2} (t)) + \cot (t) (2 t))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{1}{5} (t^{2} (- \csc^{2} (t))) + \frac{1}{5} (\cot (t) (2 t))$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $t^{2}$ and $\frac{1}{5}$ .

$\frac{t^{2}}{5} (- \csc^{2} (t)) + \frac{1}{5} (\cot (t) (2 t))$

Step 5.2.2

Combine $\frac{t^{2}}{5}$ and $\csc^{2} (t)$ .

$- \frac{t^{2} \csc^{2} (t)}{5} + \frac{1}{5} (\cot (t) (2 t))$

Step 5.2.3

Combine $\cot (t)$ and $\frac{1}{5}$ .

$- \frac{t^{2} \csc^{2} (t)}{5} + \frac{\cot (t)}{5} (2 t)$

Step 5.2.4

Combine $2$ and $\frac{\cot (t)}{5}$ .

$- \frac{t^{2} \csc^{2} (t)}{5} + \frac{2 \cot (t)}{5} t$

Step 5.2.5

Combine $\frac{2 \cot (t)}{5}$ and $t$ .

$- \frac{t^{2} \csc^{2} (t)}{5} + \frac{2 \cot (t) t}{5}$

Step 5.3

Reorder factors in $- \frac{t^{2} \csc^{2} (t)}{5} + \frac{2 \cot (t) t}{5}$ .

$- \frac{t^{2} \csc^{2} (t)}{5} + \frac{2 t \cot (t)}{5}$