Question

How do you take the derivative of `tan(sin x)`?

Answer 1

`d/(dx)tan(sinx)=cosxsec^2(sinx)`

Explanation:

For this we can use the chain rule: `d/(dx)tan(sinx)=(dsinx)/(dx)d/(dsinx)tan(sinx)=cosxd/(dsinx)tan(sinx)`

`tanx=sinx/cosx`, so `d/(dx)tanx=(cos^2x+sin^2x)/cos^2x=1/cos^2x=sec^2x`, using the quotient rule and some trigonometric identities.

Therefore we find `d/(dx)tan(sinx)=cosxsec^2(sinx)`.