Calculus Examples

cos (arccos (x))

$\cos (\arccos (x))$

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,

$\cos (\arccos (x))$ is

$x$ .

$\frac{d}{d x} [x]$

Step 2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$1$