Calculus Examples

$y = \arctan (\sqrt{4 t})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 t}$ as ${(4 t)}^{\frac{1}{2}}$ .

$y = \arctan ({(4 t)}^{\frac{1}{2}})$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} (\arctan ({(4 t)}^{\frac{1}{2}}))$

Step 3

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Factor $4$ out of $4 t$ .

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

Step 4.2

Simplify the expression.

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Step 4.2.1

Apply the product rule to $4 (t)$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Evaluate the exponent.

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

Step 4.5

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 $g (t) = 2 t^{\frac{1}{2}}$ .

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Step 4.5.1

To apply the Chain Rule, set $u$ as $2 t^{\frac{1}{2}}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Replace all occurrences of $u$ with $2 t^{\frac{1}{2}}$ .

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

Step 4.6

Simplify with factoring out.

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Step 4.6.1

Factor $2$ out of $2 t^{\frac{1}{2}}$ .

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

Step 4.6.2

Simplify the expression.

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Step 4.6.2.1

Apply the product rule to $2 (t^{\frac{1}{2}})$ .

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

Step 4.6.2.2

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

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

Step 4.6.2.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 4.6.2.3.1

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

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

Step 4.6.2.3.2

Cancel the common factor of $2$ .

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Step 4.6.2.3.2.1

Cancel the common factor.

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

Step 4.6.2.3.2.2

Rewrite the expression.

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

Step 4.7

Simplify.

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

Step 4.8

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

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

Step 4.9

Combine $2$ and $\frac{1}{1 + 4 t}$ .

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Combine the numerators over the common denominator.

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

Step 4.14

Simplify the numerator.

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Step 4.14.1

Multiply $- 1$ by $2$ .

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

Step 4.14.2

Subtract $2$ from $1$ .

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

Step 4.15

Move the negative in front of the fraction.

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

Step 4.16

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

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

Step 4.17

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

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

Step 4.18

Simplify the expression.

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Step 4.18.1

Move $2$ to the left of $1 + 4 t$ .

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

Step 4.18.2

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

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

Step 4.19

Cancel the common factor.

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

Step 4.20

Rewrite the expression.

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

Step 4.21

Simplify.

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Step 4.21.1

Apply the distributive property.

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

Step 4.21.2

Combine terms.

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Step 4.21.2.1

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

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

Step 4.21.2.2

Multiply $t$ by $t^{\frac{1}{2}}$ by adding the exponents.

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Step 4.21.2.2.1

Move $t^{\frac{1}{2}}$ .

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

Step 4.21.2.2.2

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

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Step 4.21.2.2.2.1

Raise $t$ to the power of $1$ .

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

Step 4.21.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.21.2.2.3

Write $1$ as a fraction with a common denominator.

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

Step 4.21.2.2.4

Combine the numerators over the common denominator.

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

Step 4.21.2.2.5

Add $1$ and $2$ .

$\frac{1}{t^{\frac{1}{2}} + 4 t^{\frac{3}{2}}}$

Step 4.21.3

Reorder terms.

$\frac{1}{4 t^{\frac{3}{2}} + t^{\frac{1}{2}}}$

Step 4.21.4

Simplify the denominator.

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Step 4.21.4.1

Factor $t^{\frac{1}{2}}$ out of $4 t^{\frac{3}{2}} + t^{\frac{1}{2}}$ .

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Step 4.21.4.1.1

Factor $t^{\frac{1}{2}}$ out of $4 t^{\frac{3}{2}}$ .

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

Step 4.21.4.1.2

Multiply by $1$ .

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

Step 4.21.4.1.3

Factor $t^{\frac{1}{2}}$ out of $t^{\frac{1}{2}} (4 t^{\frac{2}{2}}) + t^{\frac{1}{2}} \cdot 1$ .

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

Step 4.21.4.2

Divide $2$ by $2$ .

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

Step 4.21.4.3

Simplify.

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = \frac{1}{t^{\frac{1}{2}} (4 t + 1)}$

Step 6

Replace $y'$ with $\frac{d y}{d t}$ .

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