Calculus Examples

$\sec (\arctan (2 x))$

Step 1

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

$\sqrt{1 + {(2 x)}^{2}}$

Step 2

Apply the product rule to $2 x$ .

$\sqrt{1 + 2^{2} x^{2}}$

Step 3

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

$\sqrt{1 + 4 x^{2}}$