Calculus Examples

$2 + 5 \arctan (\frac{t}{2})$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $2 + 5 \arctan (\frac{t}{2})$ with respect to $t$ is $\frac{d}{d t} [2] + \frac{d}{d t} [5 \arctan (\frac{t}{2})]$ .

$\frac{d}{d t} [2] + \frac{d}{d t} [5 \arctan (\frac{t}{2})]$

Step 1.2

Since $2$ is constant with respect to $t$ , the derivative of $2$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [5 \arctan (\frac{t}{2})]$

Step 2

Evaluate $\frac{d}{d t} [5 \arctan (\frac{t}{2})]$ .

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Step 2.1

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

$0 + 5 \frac{d}{d t} [\arctan (\frac{t}{2})]$

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{t}{2}$ .

$0 + 5 (\frac{d}{d u} [\arctan (u)] \frac{d}{d t} [\frac{t}{2}])$

Step 2.2.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$0 + 5 (\frac{1}{1 + u^{2}} \frac{d}{d t} [\frac{t}{2}])$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{t}{2}$ .

$0 + 5 (\frac{1}{1 + {(\frac{t}{2})}^{2}} \frac{d}{d t} [\frac{t}{2}])$

Step 2.3

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

$0 + 5 (\frac{1}{1 + {(\frac{t}{2})}^{2}} (\frac{1}{2} \frac{d}{d t} [t]))$

Step 2.4

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

$0 + 5 (\frac{1}{1 + {(\frac{t}{2})}^{2}} (\frac{1}{2} \cdot 1))$

Step 2.5

Multiply $\frac{1}{2}$ by $1$ .

$0 + 5 (\frac{1}{1 + {(\frac{t}{2})}^{2}} \cdot \frac{1}{2})$

Step 2.6

Multiply $\frac{1}{1 + {(\frac{t}{2})}^{2}}$ by $\frac{1}{2}$ .

$0 + 5 \frac{1}{(1 + {(\frac{t}{2})}^{2}) \cdot 2}$

Step 2.7

Move $2$ to the left of $1 + {(\frac{t}{2})}^{2}$ .

$0 + 5 \frac{1}{2 \cdot (1 + {(\frac{t}{2})}^{2})}$

Step 2.8

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

$0 + \frac{5}{2 (1 + {(\frac{t}{2})}^{2})}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $\frac{t}{2}$ .

$0 + \frac{5}{2 (1 + \frac{t^{2}}{2^{2}})}$

Step 3.2

Apply the distributive property.

$0 + \frac{5}{2 \cdot 1 + 2 \frac{t^{2}}{2^{2}}}$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $1$ .

$0 + \frac{5}{2 + 2 \frac{t^{2}}{2^{2}}}$

Step 3.3.2

Raise $2$ to the power of $2$ .

$0 + \frac{5}{2 + 2 \frac{t^{2}}{4}}$

Step 3.3.3

Combine $2$ and $\frac{t^{2}}{4}$ .

$0 + \frac{5}{2 + \frac{2 t^{2}}{4}}$

Step 3.3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.3.4.1

Factor $2$ out of $2 t^{2}$ .

$0 + \frac{5}{2 + \frac{2 (t^{2})}{4}}$

Step 3.3.4.2

Cancel the common factors.

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Step 3.3.4.2.1

Factor $2$ out of $4$ .

$0 + \frac{5}{2 + \frac{2 t^{2}}{2 \cdot 2}}$

Step 3.3.4.2.2

Cancel the common factor.

$0 + \frac{5}{2 + \frac{2 t^{2}}{2 \cdot 2}}$

Step 3.3.4.2.3

Rewrite the expression.

$0 + \frac{5}{2 + \frac{t^{2}}{2}}$

Step 3.3.5

Add $0$ and $\frac{5}{2 + \frac{t^{2}}{2}}$ .

$\frac{5}{2 + \frac{t^{2}}{2}}$

Step 3.4

Reorder terms.

$\frac{5}{\frac{t^{2}}{2} + 2}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{\frac{t^{2}}{2} + 2 \cdot \frac{2}{2}}$

Step 3.5.2

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

$\frac{5}{\frac{t^{2}}{2} + \frac{2 \cdot 2}{2}}$

Step 3.5.3

Combine the numerators over the common denominator.

$\frac{5}{\frac{t^{2} + 2 \cdot 2}{2}}$

Step 3.5.4

Multiply $2$ by $2$ .

$\frac{5}{\frac{t^{2} + 4}{2}}$

Step 3.6

Multiply the numerator by the reciprocal of the denominator.

$5 \frac{2}{t^{2} + 4}$

Step 3.7

Multiply $5 \frac{2}{t^{2} + 4}$ .

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Step 3.7.1

Combine $5$ and $\frac{2}{t^{2} + 4}$ .

$\frac{5 \cdot 2}{t^{2} + 4}$

Step 3.7.2

Multiply $5$ by $2$ .

$\frac{10}{t^{2} + 4}$