lim as x approaches 0
(1-Cos^2(3x))/x^2
How do I derive the numerator??
derivative of 1 - cos^2 (3x)
= derivative of 1 - (cos(3x))^2
= -2cos(3x) (3)(-sin(3x)
= -3(2sin(3x)(cos(3x))
= -3sin (6x)
Is that Chain rule you used? Or..
yes, that's why I wrote it as (cos(3x))^2 , so you would see it that way
Is it a double chain?
To derive the numerator:
1. Start by applying the power rule of differentiation, which states that if you have a function of the form f(x) = x^n, then the derivative is given by f'(x) = n * x^(n-1).
2. In this case, the numerator is (1 - cos^2(3x)). We can rewrite this expression as (1 - cos^2(3x)) = sin^2(3x).
3. Now, we can differentiate sin^2(3x) using the chain rule. The chain rule states that if you have a composition of functions, you need to differentiate the outer function and then multiply it by the derivative of the inner function.
The derivative of sin^2(3x) can be found as follows:
- Differentiate the outer function: 2 * sin(3x)
- Multiply by the derivative of the inner function: 2 * sin(3x) * (d/dx) (3x)
- Simplifying the derivative of the inner function: 2 * sin(3x) * 3
- The final derivative is: 6 * sin(3x).
Therefore, the derivative of (1 - cos^2(3x)) is 6 * sin(3x).