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Nd the derivative of the function ff f(x)=(cosx)/(xⁿ)

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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.

User Chodorowicz
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