f(x)= 1/4 * (sin2x)^2
find dy/dx
d/dx (1/4 * (sin2x)^2)= (sin2xcos2x)/ 2 ?
wait no its just (sin2xcos2x)
Tom's last answer is correct
or
(1/2)sin(4x)
To find dy/dx for the given function f(x) = 1/4 * (sin2x)^2, you can use the chain rule of differentiation. The chain rule states that if you have a composition of functions, you need to differentiate the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to x.
Let's break down the function into its constituent parts.
f(x) = 1/4 * (sin2x)^2
The outer function is f(x) = x^2, and the inner function is sin2x. Let's denote the inner function as u.
So, u = sin2x
Now let's differentiate the outer function f(x) = u^2 with respect to u.
df/du = 2u
Next, let's differentiate the inner function u = sin2x with respect to x using the chain rule.
du/dx = d(sin2x)/dx = cos2x * d(2x)/dx = 2cos2x
Now, applying the chain rule, we can find the derivative dy/dx.
dy/dx = (df/du) * (du/dx) = 2u * 2cos2x = 4ucos2x
Substituting u = sin2x back into the equation, we get:
dy/dx = 4 * sin2x * cos2x
So the derivative of f(x) = 1/4 * (sin2x)^2 is dy/dx = 4 * sin2x * cos2x.