Question

What is the third derivative of tan x?

Answer 1

Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:                                                                             $\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)$

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step 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]:                                       $\displaystyle y'' = 2 \sec (x) \cdot \frac{d}{dx}[\sec (x)]$
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]:                                                                     $\displaystyle y''' = \frac{d}{dx}[2 \sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]$
6. Rewrite [Derivative Property - Multiplied Constant]:                                  $\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \frac{d}{dx}[\tan (x)]$
7. Trigonometric Differentiation:                                                                       $\displaystyle y''' = 2 \frac{d}{dx}[\sec^2 (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)$
8. Basic Power Rule [Derivative Rule - Chain Rule]:                                       $\displaystyle y''' = 2 \big( 2 \sec (x) \big) \frac{d}{dx}[\sec (x)] \tan (x) + 2 \sec^2 (x) \cdot \sec^2 (x)$
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