Calculus Examples

$y = \arccos (\frac{11}{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \arccos (x)$ and $g (x) = \frac{11}{x}$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u$ as $\frac{11}{x}$ .

$\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [\frac{11}{x}]$

Step 1.2

The derivative of $\arccos (u)$ with respect to $u$ is $- \frac{1}{\sqrt{1 - u^{2}}}$ .

$- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [\frac{11}{x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{11}{x}$ .

$- \frac{1}{\sqrt{1 - {(\frac{11}{x})}^{2}}} \frac{d}{d x} [\frac{11}{x}]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since $11$ is constant with respect to $x$ , the derivative of $\frac{11}{x}$ with respect to $x$ is $11 \frac{d}{d x} [\frac{1}{x}]$ .

$- \frac{1}{\sqrt{1 - {(\frac{11}{x})}^{2}}} (11 \frac{d}{d x} [\frac{1}{x}])$

Step 2.2

Combine fractions.

Tap for more steps...

Step 2.2.1

Multiply $11$ by $- 1$ .

$- 11 \frac{1}{\sqrt{1 - {(\frac{11}{x})}^{2}}} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.2

Combine $- 11$ and $\frac{1}{\sqrt{1 - {(\frac{11}{x})}^{2}}}$ .

$\frac{- 11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.3

Simplify the expression.

Tap for more steps...

Step 2.2.3.1

Move the negative in front of the fraction.

$- \frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} \frac{d}{d x} [\frac{1}{x}]$

Step 2.2.3.2

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

$- \frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} \frac{d}{d x} [x^{- 1}]$

Step 2.3

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

$- \frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} (- x^{- 2})$

Step 2.4

Combine fractions.

Tap for more steps...

Step 2.4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} x^{- 2}$

Step 2.4.2

Multiply $\frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}}$ by $1$ .

$\frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}} x^{- 2}$

Step 2.4.3

Combine $\frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}}}$ and $x^{- 2}$ .

$\frac{11 x^{- 2}}{\sqrt{1 - {(\frac{11}{x})}^{2}}}$

Step 2.4.4

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

$\frac{11}{\sqrt{1 - {(\frac{11}{x})}^{2}} x^{2}}$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the product rule to $\frac{11}{x}$ .

$\frac{11}{\sqrt{1 - \frac{11^{2}}{x^{2}}} x^{2}}$

Step 3.2

Raise $11$ to the power of $2$ .

$\frac{11}{\sqrt{1 - \frac{121}{x^{2}}} x^{2}}$

Step 3.3

Reorder terms.

$\frac{11}{x^{2} \sqrt{1 - \frac{121}{x^{2}}}}$