Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
Let y = (x + sec x)(x - tan x)
Using the product rule:
dy/dx = d/dx(x + sec x) · (x - tan x) + (x + sec x) · d/dx(x - tan x)
= (1 + sec x tan x)(x - tan x) + (x + sec x)(1 - sec² x)
Explanation
This is a direct application of the product rule for differentiation. The context shows the quotient rule and product rule being applied to similar functions. The derivative of sec x is sec x tan x, and the derivative of tan x is sec² x (mentioned in the context). For a 1-mark question, showing the product rule application with correct derivatives is sufficient.
Solution Steps
Step 1: Identify the function as a product of two functions: u = (x + sec x) and v = (x - tan x)
Step 2: Apply the product rule: dy/dx = u'v + uv'
Step 3: Find u' = d/dx(x + sec x) = 1 + sec x tan x
Step 4: Find v' = d/dx(x - tan x) = 1 - sec² x
Step 5: Substitute: dy/dx = (1 + sec x tan x)(x - tan x) + (x + sec x)(1 - sec² x)