find f'(x)

f(x)=cos(2x)tan(2x)/2√x

To find the derivative of f(x), which is denoted as f'(x), we can use the chain rule and the product rule. Here's how to solve it step by step:

Step 1: Rewrite the function f(x) to simplify the expression:
f(x) = (cos(2x) * tan(2x)) / (2√x)
= (cos(2x) * sin(2x) / cos(2x)) / (2√x)
= sin(2x) / (2√x)

Step 2: Apply the quotient rule to differentiate the function f(x):
The quotient rule states that if you have a function u(x) / v(x), then its derivative f'(x) is given by
f'(x) = (u'(x) * v(x) - u(x) * v'(x)) / (v(x))^2

Let's apply this rule to our function:

In our case, u(x) = sin(2x) and v(x) = 2√x.
- Find the derivative of u(x):
u'(x) = d/dx(sin(2x)) = 2 * cos(2x)

- Find the derivative of v(x):
v'(x) = d/dx(2√x) = 2 * (1 / (2√x)) * (1/2)x^(-1/2) = 1 / √x

Step 3: Substitute the derivatives into the quotient rule formula:
f'(x) = [(2 * cos(2x) * 2√x) - (sin(2x) * (1 / √x))] / (2√x)^2
= (4√x * cos(2x) - sin(2x) / √x) / (4x)

Finally, simplifying the expression, the resulting derivative of f(x) is:
f'(x) = (4√x * cos(2x) - sin(2x)) / (4x√x)

So, the derivative of f(x) is (4√x * cos(2x) - sin(2x)) / (4x√x).