Question

derivative by first principle logsecx²​

Answer 1

Answer:

$\displaystyle \frac{d}{dx}[\log \big( \sec (x^2) \big)] = \frac{2x \tan x^2}{\ln 10}$

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

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

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

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

Unit: Differentiation