What is the derivative of cos(cos(sin(x))) and what the derivative of tan(sin(2x))

please solve this out because i cant see it especially the one with the half angle identity i dont know when to use the chain rule or when to actually to use the product,sum,quotient , and subtraction rule,
Please help!

Start by working on the outside to the inside. First take the derivative of just the cos on the outside:
-sin(cos(sin(x)))
Now multiple this by the derivative of what's in the parenthesis cos(sin(x)):
-sin(cos(sin(x))) *-sin(sin(x))
Finally multiply this by the derivative of what's in the parenthesis sin(x):
-sin(cos(sin(x))) *-sin(sin(x)) *cos(x)
Your final answer will be:
sin(cos(sin(x)))sin(sin(x))cos(x)

Use the same process:
tan(sin(2x))
sec^2(sin(2x))
sec^2(sin(2x)) *cos(2x)
sec^2(sin(2x)) *cos(2x) *2
2sec^2(sin(2x))cos(2x)

Hope this helps!

No problem at all.

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cos(cos(sin(x))), let y=cos(sin(x))

use chain rule:

-sin(y)*y'

y' = -sin(sin(x))cos(x)

-sin(cos(sin(x)))*-sin(sin(x))cos(x)

tan(sin(2x))

2sec^2(sin(2x))*cos(2x)