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What is the derivative of secx​

User Railmisaka
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Answer:


\displaystyle (d)/(dx)[\sec x] = \sec x \tan x

General Formulas and Concepts:

Pre-Calculus

  • Trigonometric Identities

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

*Note:

This is a known trigonometric derivative.

Step 1: Define

Identify


\displaystyle y = \sec x

Step 2: Differentiate

  1. Rewrite [Trigonometric Identities]:
    \displaystyle y = (1)/(\cos x)
  2. Derivative Rule [Quotient Rule]:
    \displaystyle y' = ((1)' \cos x - 1(\cos x)')/(\cos^2 x)
  3. Basic Power Rule:
    \displaystyle y' = ((0) \cos x - 1(\cos x)')/(\cos^2 x)
  4. Trigonometric Differentiation:
    \displaystyle y' = ((0) \cos x + 1(\sin x))/(\cos^2 x)
  5. Simplify:
    \displaystyle y' = (\sin x)/(\cos^2 x)
  6. Rewrite:
    \displaystyle y' = (\sin x)/(\cos x) \cdot (1)/(\cos x)
  7. Rewrite [Trigonometric Identities]:
    \displaystyle y' = \tan x \sec x

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

Unit: Differentiation

User Oswaldo Ferreira
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