How to differentiate cos(2x)/x^½

To differentiate the expression cos(2x)/x^(1/2), one must apply the quotient rule of differentiation, which is essential when dealing with a function that is the ratio of two other functions. The quotient rule states that if you have a function in the form of f(x)/g(x), the derivative can be calculated using the formula (f'(x)g(x) – f(x)g'(x)) / (g(x))^2. In this case, let f(x) be cos(2x) and g(x) be x^(1/2). The first step involves finding the derivatives of both f(x) and g(x).

The derivative of f(x) = cos(2x) can be determined using the chain rule. The derivative f'(x) is -2sin(2x), as the derivative of cos(u) is -sin(u) multiplied by the derivative of the inner function, which in this case is 2. For g(x) = x^(1/2), the derivative g'(x) is (1/2)x^(-1/2), which simplifies to 1/(2√x). With these derivatives calculated, one can now substitute them back into the quotient rule formula to find the overall derivative of the original expression.

Substituting the derivatives into the quotient rule formula yields the following expression: [(-2sin(2x))(x^(1/2)) – (cos(2x))(1/(2√x))] / (x^(1/2))^2. This can be further simplified to express the derivative in a more manageable form. The denominator simplifies to x, while the numerator can be rearranged to combine the terms effectively. Thus, the final result provides a clear representation of the derivative of the function cos(2x)/x^(1/2), which can be utilised for further analysis or application in calculus.

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