To find the derivative of cos^2(4x), you can use the chain rule. The chain rule states that if you have a composite function, f(g(x)), then the derivative of the composite function is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Let's break down the function cos^2(4x) into its components:
- The outer function is cos^2(u), where u = 4x.
- The inner function is u = 4x.
We can find the derivatives of the outer and inner functions separately.
1. Derivative of the outer function:
To find the derivative of cos^2(u), we treat it as (cos(u))^2 and apply the chain rule.
Let y = cos(u).
Therefore, y^2 = (cos(u))^2.
Now, let's find the derivative of y^2 with respect to y, which is 2y.
Therefore, the derivative of cos^2(u) with respect to u is 2cos(u).
2. Derivative of the inner function:
To find the derivative of u = 4x, we treat it as a simple linear equation.
The derivative of 4x with respect to x is 4.
Now, we can apply the chain rule by multiplying the derivatives of the outer and inner functions:
The derivative of cos^2(4x) = (2cos(u)) * (4).
Substituting u = 4x back in, we get:
The derivative of cos^2(4x) = (2cos(4x)) * (4).
Therefore, the final derivative of cos^2(4x) is 8cos(4x).