Thread: Differentiation of Trigonometric fuctions in terms of Exoponential Functions

Having a bit of difficulty with some calculus where I have to differentiate sin,cos,tan, sinh,cosh and tanh when they are in terms of their exponential functions, in accordance with DeMoivre's Theorem

In particular, d/dt(tan t)
i have gone down this route:
= d/dt(sint/cost) = d/dt {(exp it - exp -it) / i(exp it + exp -it)}
I am unsure about the next step, as it seems everything I have tried gives me a lot of mess!
I have tried multipplying top and bottom by exp it and differentiated to get something in terms of exp 4it
This doesn't help at all!
I know I'm aiming for sec^2 x... so looking for 1/ (exp it + exp -it)
but i can't get there

also finding it difficult to do d/dt (tanh t), but i have successfully completed sin t, cos t, sinh t and cosh t
Please could someone point in the right direction with the question mentioned above? I'm going round in circles!!

Cheers

Originally Posted by natscitmf

In particular, d/dt(tan t)
i have gone down this route:
= d/dt(sint/cost) = d/dt {(exp it - exp -it) / i(exp it + exp -it)}
I am unsure about the next step, as it seems everything I have tried gives me a lot of mess!

No messier than this, surely...

Originally Posted by natscitmf

I know I'm aiming for sec^2 x... so looking for 1/ (exp it + exp -it)

I make that 4/ (exp it + exp -it)^2 ... Notice the bottom row, above, already has the denominator common to both parts of the sum, and the numerators simplify to 2 + 2.

By the way,



... is the chain rule, here wrapped inside the legs-uncrossed version of ...



... the product rule... the whole thing hopefully no more fearsome than your common or garden quotient rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which subject to the chain rule).


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