How do you find the derivative of #y = sin(tan 2x)#?

The chain rule must be applied, which implies that

Regarding you, we have:

These functions, when entered into the original formula, yield:

After dividing the three, you obtain

To find the derivative of ( y = \sin(\tan(2x)) ), you can use the chain rule. Here's the process:

1. Identify the outer function as (\sin(u)) and the inner function as (\tan(2x)).
2. Find the derivative of the outer function with respect to its inner function: (\frac{d}{du}[\sin(u)] = \cos(u)).
3. Find the derivative of the inner function with respect to (x): (\frac{d}{dx}[\tan(2x)] = 2\sec^2(2x)).
4. Apply the chain rule: Multiply the derivative of the outer function by the derivative of the inner function.
5. Substitute the inner function back into the result.

Therefore, the derivative of ( y = \sin(\tan(2x)) ) with respect to (x) is:

[ y' = \cos(\tan(2x)) \cdot 2\sec^2(2x) ]