Question

What is the third derivative of tan x?

Answer 1

`\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = \tan x`

Step 2: Differentiate

1. Trigonometric Differentiation:
   `\displaystyle y' = \sec^2 (x)`
2. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y'' = 2 \sec (x) \cdot (d)/(dx)[\sec (x)]`
3. Trigonometric Differentiation:
   `\displaystyle y'' = 2 \sec (x) \cdot \sec (x) \tan (x)`
4. Simplify:
   `\displaystyle y'' = 2 \sec^2 (x) \tan (x)`
5. Derivative Rule [Product Rule]:
   `\displaystyle y''' = (d)/(dx)[2 \sec^2 (x)] \tan (x) + 2 \sec^2 (x) (d)/(dx)[\tan (x)]`
6. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle y''' = 2 (d)/(dx)[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) (d)/(dx)[\tan (x)]`
7. Trigonometric Differentiation:
   `\displaystyle y''' = 2 (d)/(dx)[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)`
8. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y''' = 2 \big( 2 \sec (x) \big) (d)/(dx)[\sec (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)`
9. Trigonometric Differentiation:
   `\displaystyle y''' = 2 \big( 2 \sec (x) \big) \big( \sec (x) \tan (x) \big) \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)`
10. Simplify:
    `\displaystyle y''' = 4 \sec^2 (x) \tan^2 (x) + 2 \sec^4 (x)`
11. Factor:
    `\displaystyle y''' = 2 \sec^2 (x) \bigg( 2\tan^2 (x) + \sec^2 (x) \bigg)`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation