Final answer:
The derivative of the function f(x) = (cosx) / (x^n) is found using the quotient rule, resulting in a combination of trigonometric functions and algebraic terms.
Step-by-step explanation:
To find the derivative of the function f(x) = (cosx) / (x^n), we need to apply the quotient rule since this is a division of two functions. The quotient rule is given by:
d/dx [u(x) / v(x)] = [v(x) * d/dx u(x) - u(x) * d/dx v(x)] / [v(x)]^2
Let u(x) = cos(x) and v(x) = x^n. Then the derivatives are d/dx u(x) = -sin(x) and d/dx v(x) = n*x^(n-1).
Applying the quotient rule:
d/dx f(x) = [x^n * (-sin(x)) - cos(x) * n*x^(n-1)] / (x^n)^2
Simplifying the expression:
d/dx f(x) = [-x^n * sin(x) - n*cos(x) * x^(n-1)] / x^(2n)
d/dx f(x) = [-sin(x) - n*cos(x) / x] / x^n
The final derivative is a combination of trigonometric functions and algebraic expressions, which we obtained by methodically following the differentiation rules.