Calculus Examples

$y = \arctan (\sqrt{3 x})$

Step 1

Simplify with factoring out.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\arctan ({(3 x)}^{\frac{1}{2}})]$

Step 1.2

Factor $3$ out of $3 x$ .

$\frac{d}{d x} [\arctan ({(3 (x))}^{\frac{1}{2}})]$

Step 1.3

Apply the product rule to $3 (x)$ .

$\frac{d}{d x} [\arctan (3^{\frac{1}{2}} x^{\frac{1}{2}})]$

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 $g (x) = 3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 2.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [3^{\frac{1}{2}} x^{\frac{1}{2}}]$

Step 2.3

Replace all occurrences of $u$ with $3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 3

Differentiate.

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Step 3.1

Since $3^{\frac{1}{2}}$ is constant with respect to $x$ , the derivative of $3^{\frac{1}{2}} x^{\frac{1}{2}}$ with respect to $x$ is $3^{\frac{1}{2}} \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

Move the negative in front of the fraction.

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

Step 9

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

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

Step 10

Multiply $\frac{3^{\frac{1}{2}}}{1 + {(3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}}$ by $\frac{x^{- \frac{1}{2}}}{2}$ .

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

Step 11

Simplify the expression.

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Step 11.1

Move $2$ to the left of $1 + {(3^{\frac{1}{2}} x^{\frac{1}{2}})}^{2}$ .

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

Step 11.2

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

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

Step 12

Simplify.

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Step 12.1

Apply the product rule to $3^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Apply the distributive property.

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

Step 12.4

Combine terms.

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Step 12.4.1

Multiply $2$ by $1$ .

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

Step 12.4.2

Multiply the exponents in ${(3^{\frac{1}{2}})}^{2}$ .

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Step 12.4.2.1

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

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

Step 12.4.2.2

Cancel the common factor of $2$ .

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Step 12.4.2.2.1

Cancel the common factor.

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

Step 12.4.2.2.2

Rewrite the expression.

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

Step 12.4.3

Evaluate the exponent.

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

Step 12.4.4

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

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Step 12.4.4.1

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

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

Step 12.4.4.2

Cancel the common factor of $2$ .

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Step 12.4.4.2.1

Cancel the common factor.

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

Step 12.4.4.2.2

Rewrite the expression.

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

Step 12.4.5

Simplify.

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

Step 12.4.6

Multiply $3$ by $2$ .

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 x \cdot x^{\frac{1}{2}}}$

Step 12.4.7

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

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Step 12.4.7.1

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

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 (x^{\frac{1}{2}} x)}$

Step 12.4.7.2

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

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Step 12.4.7.2.1

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

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 (x^{\frac{1}{2}} x^{1})}$

Step 12.4.7.2.2

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

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 x^{\frac{1}{2} + 1}}$

Step 12.4.7.3

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

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 x^{\frac{1}{2} + \frac{2}{2}}}$

Step 12.4.7.4

Combine the numerators over the common denominator.

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 x^{\frac{1 + 2}{2}}}$

Step 12.4.7.5

Add $1$ and $2$ .

$\frac{3^{\frac{1}{2}}}{2 x^{\frac{1}{2}} + 6 x^{\frac{3}{2}}}$

Step 12.5

Reorder terms.

$\frac{3^{\frac{1}{2}}}{6 x^{\frac{3}{2}} + 2 x^{\frac{1}{2}}}$

Step 12.6

Simplify the denominator.

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Step 12.6.1

Factor $2 x^{\frac{1}{2}}$ out of $6 x^{\frac{3}{2}} + 2 x^{\frac{1}{2}}$ .

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Step 12.6.1.1

Factor $2 x^{\frac{1}{2}}$ out of $6 x^{\frac{3}{2}}$ .

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

Step 12.6.1.2

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

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

Step 12.6.1.3

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

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

Step 12.6.2

Divide $2$ by $2$ .

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

Step 12.6.3

Simplify.

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