Final answer:
The derivative of the given function is f'(x) = 10cos(x)sec(x) - 10sec^2(x)
Step-by-step explanation:
To find the derivative of the function f(x) = [\frac{sin(x) * 10sec(x)}{1 + xtan(x)}], we can use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = \frac{g(x)}{h(x)}, then the derivative f'(x) can be found using the formula:
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
Applying the quotient rule to the given function, we have:
f'(x) = \frac{(sin(x) * 10(sec(x))^2) - (10sec(x) * (1 + xtan(x)))}{(1 + xtan(x))^2}
Simplifying the expression further:
f'(x) = 10cos(x)sec(x) - 10sec^2(x)