Differentiate sin(ax), cos(ax) and tan(ax) from first principles.
The Attempt at a Solution
I have used first principles to differentiate the three expressions and have been successful until I encountered limits of some expressions in the process.
I need to find the limit as Δx tends to 0 of the following expressions.
2. [cos(Δx) - 1]/Δx
I know some spooky proofs which use the fact that for small Δx, sin(aΔx) ≈ Δx, cos(aΔx) ≈ 1- (aΔx)2/2 and tan(aΔx) ≈ Δx. They do give the right answers and I have been told these methods would give me full marks in the exam (me being a physics student and all that crap!), but I would appreciate it if you give a full rigorous proof of the three limits. (Armed with those, the original problem is just a piece of cake.)