Derivative of tan(2x+3) using first principle

The derivative is given by the limit



You have




Use the angle sum identity for tangent. I don't remember it off the top of my head, but I do remember the ones for (co)sine.



By this identity, you have



So in the limit you get







The first two limits are both 1, and the single term in the last limit approaches 0 as , so you're left with



which agrees with the result you get from applying the chain rule.