Question

What is the derivative of sec(x)^2

Answer 1

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!

Answer 2

`\displaystyle (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 (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

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 (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