Calculus Examples

$f (x) = \sqrt{arccot (6 x)}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{arccot (6 x)}$ as ${arccot (6 x)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{arccot (6 x)}^{\frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = arccot (6 x)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $arccot (6 x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [arccot (6 x)]$

Step 2.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [arccot (6 x)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $arccot (6 x)$ .

$\frac{1}{2} {arccot (6 x)}^{\frac{1}{2} - 1} \frac{d}{d x} [arccot (6 x)]$

Step 3

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

$\frac{1}{2} {arccot (6 x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 4

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

$\frac{1}{2} {arccot (6 x)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} {arccot (6 x)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {arccot (6 x)}^{\frac{1 - 2}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {arccot (6 x)}^{\frac{- 1}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$\frac{1}{2} {arccot (6 x)}^{- \frac{1}{2}} \frac{d}{d x} [arccot (6 x)]$

Step 7.2

Combine $\frac{1}{2}$ and ${arccot (6 x)}^{- \frac{1}{2}}$ .

$\frac{{arccot (6 x)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [arccot (6 x)]$

Step 7.3

Move ${arccot (6 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} \frac{d}{d x} [arccot (6 x)]$

Step 8

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = arccot (x)$ and $g (x) = 6 x$ .

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Step 8.1

To apply the Chain Rule, set $u_{2}$ as $6 x$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [arccot (u_{2})] \frac{d}{d x} [6 x])$

Step 8.2

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

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (- \frac{1}{1 + {u_{2}}^{2}} \frac{d}{d x} [6 x])$

Step 8.3

Replace all occurrences of $u_{2}$ with $6 x$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (- \frac{1}{1 + {(6 x)}^{2}} \frac{d}{d x} [6 x])$

Step 9

Differentiate using the Constant Multiple Rule.

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Step 9.1

Factor $6$ out of $6 x$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (- \frac{1}{1 + {(6 (x))}^{2}} \frac{d}{d x} [6 x])$

Step 9.2

Combine fractions.

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Step 9.2.1

Simplify the expression.

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Step 9.2.1.1

Apply the product rule to $6 (x)$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (- \frac{1}{1 + 6^{2} x^{2}} \frac{d}{d x} [6 x])$

Step 9.2.1.2

Raise $6$ to the power of $2$ .

$\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}} (- \frac{1}{1 + 36 x^{2}} \frac{d}{d x} [6 x])$

Step 9.2.2

Multiply $\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}}}$ by $\frac{1}{1 + 36 x^{2}}$ .

$- \frac{1}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [6 x]$

Step 9.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$- \frac{1}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} (6 \frac{d}{d x} [x])$

Step 9.4

Simplify terms.

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Step 9.4.1

Multiply $6$ by $- 1$ .

$- 6 \frac{1}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [x]$

Step 9.4.2

Combine $- 6$ and $\frac{1}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})}$ .

$\frac{- 6}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [x]$

Step 9.4.3

Factor $2$ out of $- 6$ .

$\frac{2 \cdot - 3}{2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [x]$

Step 10

Cancel the common factors.

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Step 10.1

Factor $2$ out of $2 {arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})$ .

$\frac{2 \cdot - 3}{2 ({arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2}))} \frac{d}{d x} [x]$

Step 10.2

Cancel the common factor.

$\frac{2 \cdot - 3}{2 ({arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2}))} \frac{d}{d x} [x]$

Step 10.3

Rewrite the expression.

$\frac{- 3}{{arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [x]$

Step 11

Move the negative in front of the fraction.

$- \frac{3}{{arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \frac{d}{d x} [x]$

Step 12

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

$- \frac{3}{{arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})} \cdot 1$

Step 13

Multiply $- 1$ by $1$ .

$- \frac{3}{{arccot (6 x)}^{\frac{1}{2}} (1 + 36 x^{2})}$