Calculus Examples

$y = \tan^{-1} (\sqrt{4 t})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \tan^{-1} (t)$ and $g (t) = \sqrt{4 t}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\sqrt{4 t}$ .

$\frac{d}{d u} [\tan^{-1} (u)] \frac{d}{d t} [\sqrt{4 t}]$

Step 1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d t} [\sqrt{4 t}]$

Step 1.3

Replace all occurrences of $u$ with $\sqrt{4 t}$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \frac{d}{d t} [\sqrt{4 t}]$

Step 2

Rewrite $\frac{d}{d t} [\sqrt{4 t}]$ as $\frac{d}{d t} [2 \sqrt{t}]$ .

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Step 2.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \frac{d}{d t} [\sqrt{2^{2} t}]$

Step 2.2

Pull terms out from under the radical.

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \frac{d}{d t} [2 \sqrt{t}]$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \frac{d}{d t} [2 t^{\frac{1}{2}}]$

Step 4

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

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 \frac{d}{d t} [t^{\frac{1}{2}}])$

Step 5

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

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 7

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

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 8

Combine the numerators over the common denominator.

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}))$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{1 - 2}{2}}))$

Step 9.2

Subtract $2$ from $1$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{\frac{- 1}{2}}))$

Step 10

Move the negative in front of the fraction.

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 (\frac{1}{2} t^{- \frac{1}{2}}))$

Step 11

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

$\frac{1}{1 + {\sqrt{4 t}}^{2}} (2 \frac{t^{- \frac{1}{2}}}{2})$

Step 12

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

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \cdot \frac{2 t^{- \frac{1}{2}}}{2}$

Step 13

Move $t^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \cdot \frac{2}{2 t^{\frac{1}{2}}}$

Step 14

Cancel the common factor.

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \cdot \frac{2}{2 t^{\frac{1}{2}}}$

Step 15

Rewrite the expression.

$\frac{1}{1 + {\sqrt{4 t}}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16

Simplify.

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Step 16.1

Combine terms.

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Step 16.1.1

Rewrite $\frac{1}{1 + {\sqrt{4 t}}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$ as $\frac{1}{1 + {(2 \sqrt{t})}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$ .

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Step 16.1.1.1

Rewrite $4$ as $2^{2}$ .

$\frac{1}{1 + {\sqrt{2^{2} t}}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.1.2

Pull terms out from under the radical.

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

Step 16.1.2

Factor $2$ out of $2 \sqrt{t}$ .

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

Step 16.1.3

Apply the product rule to $2 (\sqrt{t})$ .

$\frac{1}{1 + 2^{2} {\sqrt{t}}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.4

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

$\frac{1}{1 + 4 {\sqrt{t}}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5

Rewrite ${\sqrt{t}}^{2}$ as $t$ .

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Step 16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{1}{1 + 4 {(t^{\frac{1}{2}})}^{2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{1 + 4 t^{\frac{1}{2} \cdot 2}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5.3

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

$\frac{1}{1 + 4 t^{\frac{2}{2}}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5.4

Cancel the common factor of $2$ .

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Step 16.1.5.4.1

Cancel the common factor.

$\frac{1}{1 + 4 t^{\frac{2}{2}}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5.4.2

Rewrite the expression.

$\frac{1}{1 + 4 t^{1}} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.5.5

Simplify.

$\frac{1}{1 + 4 t} \cdot \frac{1}{t^{\frac{1}{2}}}$

Step 16.1.6

Multiply $\frac{1}{1 + 4 t}$ by $\frac{1}{t^{\frac{1}{2}}}$ .

$\frac{1}{(1 + 4 t) t^{\frac{1}{2}}}$

Step 16.2

Reorder terms.

$\frac{1}{t^{\frac{1}{2}} (1 + 4 t)}$