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):
(1 - sin x)(x - tan x) + (x + cos x)(1 - sec² x)
Explanation
This question requires finding the derivative of a product of two functions. The student must apply the product rule of differentiation. From the context, we know that the derivative of tan x is sec² x (mentioned in Example 17 and used in the solution for cot x). The derivative of cos x is -sin x, and the derivative of x is 1.
Solution Steps
Step 1: Let u = x + cos x and v = x - tan x. Apply the product rule: d/dx(uv) = u'v + uv'
Step 2: Find u' = d/dx(x + cos x) = 1 - sin x
Step 3: Find v' = d/dx(x - tan x) = 1 - sec² x
Step 4: Apply product rule: (1 - sin x)(x - tan x) + (x + cos x)(1 - sec² x)