Final answer:
The derivative of f(x) = cosx(sec²x cot²x - csc²x cos²x) is found by applying product and chain rules of differentiation along with trigonometric identities.
Step-by-step explanation:
To calculate the derivative of the function f(x) = cosx(sec²x cot²x - csc²x cos²x) using the definition, we need to apply the product rule and the chain rule of differentiation, as well as trigonometric identities.
First, we recognize that the derivative of cos(x) is -sin(x). Then, we address the other parts of the function through the use of trigonometric identities. Recall that sec(x) = 1/cos(x), csc(x) = 1/sin(x), cot(x) = cos(x)/sin(x), and the derivatives sec'(x) = sec(x)tan(x), csc'(x) = -csc(x)cot(x), cot'(x) = -csc²(x).
Applying the product rule, we find the derivative of the second part of the function (sec²x cot²x - csc²x cos²x) involves taking the derivative of sec²x cot²x and -csc²x cos²x separately and summing them up.
The derivative of the whole function is the derivative of cos(x) multiplied by the original second part plus the original cos(x) multiplied by the derivative of the second part. To finalize, we simplify the expression using trigonometric identities as needed.