Calculus Examples

$y = \arctan (\sqrt{x})$

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arctan (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Move the negative in front of the fraction.

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

Step 11

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

$\frac{1}{1 + x} \cdot \frac{x^{- \frac{1}{2}}}{2}$

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Move $2$ to the left of $1 + x$ .

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

Step 13.2

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

$\frac{1}{2 (1 + x) x^{\frac{1}{2}}}$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

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

Step 14.2

Apply the distributive property.

$\frac{1}{2 \cdot 1 x^{\frac{1}{2}} + 2 x \cdot x^{\frac{1}{2}}}$

Step 14.3

Combine terms.

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Step 14.3.1

Multiply $2$ by $1$ .

$\frac{1}{2 x^{\frac{1}{2}} + 2 x \cdot x^{\frac{1}{2}}}$

Step 14.3.2

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

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Step 14.3.2.1

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

$\frac{1}{2 x^{\frac{1}{2}} + 2 (x^{\frac{1}{2}} x)}$

Step 14.3.2.2

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

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Step 14.3.2.2.1

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

$\frac{1}{2 x^{\frac{1}{2}} + 2 (x^{\frac{1}{2}} x^{1})}$

Step 14.3.2.2.2

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

$\frac{1}{2 x^{\frac{1}{2}} + 2 x^{\frac{1}{2} + 1}}$

Step 14.3.2.3

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

$\frac{1}{2 x^{\frac{1}{2}} + 2 x^{\frac{1}{2} + \frac{2}{2}}}$

Step 14.3.2.4

Combine the numerators over the common denominator.

$\frac{1}{2 x^{\frac{1}{2}} + 2 x^{\frac{1 + 2}{2}}}$

Step 14.3.2.5

Add $1$ and $2$ .

$\frac{1}{2 x^{\frac{1}{2}} + 2 x^{\frac{3}{2}}}$

Step 14.4

Simplify the denominator.

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Step 14.4.1

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

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Step 14.4.1.1

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

$\frac{1}{2 x^{\frac{1}{2}} (1) + 2 x^{\frac{3}{2}}}$

Step 14.4.1.2

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

$\frac{1}{2 x^{\frac{1}{2}} (1) + 2 x^{\frac{1}{2}} (x^{\frac{2}{2}})}$

Step 14.4.1.3

Factor $2 x^{\frac{1}{2}}$ out of $2 x^{\frac{1}{2}} (1) + 2 x^{\frac{1}{2}} (x^{\frac{2}{2}})$ .

$\frac{1}{2 x^{\frac{1}{2}} (1 + x^{\frac{2}{2}})}$

Step 14.4.2

Divide $2$ by $2$ .

$\frac{1}{2 x^{\frac{1}{2}} (1 + x^{1})}$

Step 14.4.3

Simplify.

$\frac{1}{2 x^{\frac{1}{2}} (1 + x)}$