Calculus Examples

$f (x) = \sin (\sec (\arccos (x)))$

Step 1

Draw a triangle in the plane with vertices $(x, \sqrt{1^{2} - x^{2}})$ , $(x, 0)$ , and the origin. Then $\arccos (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(x, \sqrt{1^{2} - x^{2}})$ . Therefore, $\sec (\arccos (x))$ is $\frac{1}{x}$ .

$\frac{d}{d x} [\sin (\frac{1}{x})]$

Step 2

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

Tap for more steps...

Step 2.1

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

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{1}{x}]$

Step 2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [\frac{1}{x}]$

Step 2.3

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

$\cos (\frac{1}{x}) \frac{d}{d x} [\frac{1}{x}]$

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.2

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

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