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Derivative by first principle logsecx²​

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Answer:


\displaystyle (d)/(dx)[\log \big( \sec (x^2) \big)] = (2x \tan x^2)/(\ln 10)

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify


\displaystyle y = \log \big( \sec (x^2) \big)

Step 2: Differentiate

  1. Logarithmic Differentiation [Derivative Rule - Chain Rule]:
    \displaystyle y' = ((\sec x^2)')/(\ln (10) \sec x^2)
  2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
    \displaystyle y' = (\sec x^2 \tan x^2 (x^2)')/(\ln (10) \sec x^2)
  3. Simplify:
    \displaystyle y' = (\tan x^2 (x^2)')/(\ln 10)
  4. Basic Power Rule:
    \displaystyle y' = (2x \tan x^2)/(\ln 10)

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

Unit: Differentiation

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