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Verified Answer :

Zinethar

This is also written as sec^2(x)
Using chain rule, we can separate this into 2sec(x)d/dx(sec(x))
Using trig derivative identities, we know that d/dx(sec(x))=tan(x)sec(x)
Therefore, we have 2sec(x)tan(x)sec(x), which simplifies down to 2tan(x)sec^2(x).  

Hope this helps!

Space

Answer:

\(\displaystyle \frac{dy}{dx} = 2 \tan (x) \sec^2 (x)\)

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

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

Step 2: Differentiate

  1. Basic Power Rule [Derivative Rule - Chain Rule]:                                       \(\displaystyle y' = 2 \sec (x) \cdot \frac{d}{dx}[\sec (x)]\)
  2. Trigonometric Differentiation:                                                                       \(\displaystyle y' = 2 \sec (x) \cdot \sec (x) \tan (x)\)
  3. Simplify:                                                                                                         \(\displaystyle y' = 2 \sec^2 (x) \tan (x)\)

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

Unit: Differentiation

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