Calculus Examples

$y = \frac{1}{arccot (x)}$

Step 1

Rewrite $\frac{1}{arccot (x)}$ as ${arccot (x)}^{- 1}$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $arccot (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $arccot (x)$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply ${arccot (x)}^{- 2}$ by $1$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply ${arccot (x)}^{2}$ by $1$ .

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

Step 5.3

Factor ${arccot (x)}^{2}$ out of ${arccot (x)}^{2} + x^{2} {arccot (x)}^{2}$ .

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Step 5.3.1

Multiply by $1$ .

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

Step 5.3.2

Factor ${arccot (x)}^{2}$ out of $x^{2} {arccot (x)}^{2}$ .

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

Step 5.3.3

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

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