Respuesta :

Space

Answer:

[tex]\displaystyle \frac{d}{dx}[\tan x] = \sec^2 x[/tex]

General Formulas and Concepts:

Pre-Calculus

  • Trigonometric Identities

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Rule [Quotient Rule]:                                                                           [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]

Step-by-step explanation:

*Note:

This is a known trigonometric derivative.

Step 1: Define

Identify

[tex]\displaystyle y = \tan x[/tex]

Step 2: Differentiate

  1. Rewrite [Trigonometric Identities]:                                                               [tex]\displaystyle y = \frac{\sin x}{\cos x}[/tex]
  2. Derivative Rule [Quotient Rule]:                                                                   [tex]\displaystyle y' = \frac{(\sin x)'\cos x - \sin x(\cos x)'}{\cos^2 x}[/tex]
  3. Trigonometric Differentiation:                                                                      [tex]\displaystyle y' = \frac{\cos^2 x + \sin^2 x}{\cos^2 x}[/tex]
  4. Rewrite [Trigonometric Identities]:                                                               [tex]\displaystyle y' = \frac{1}{\cos^2 x}[/tex]
  5. Rewrite [Trigonometric Identities]:                                                               [tex]\displaystyle y' = \sec^2 x[/tex]

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

Unit: Differentiation