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Can someone please check this? (differentiation)

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Original post by nuodai
It's all essentially the same thing. For the first one you pretty much already did it with this one; let y=tn(x)y=t_n(x), and then ln⁡y=tn−1(x)ln⁡x\ln y = t_{n-1}(x)\ln x. [Remember you just need to find a recursive formula in terms of tn,tn−1′,tn−1t_n, t'_{n-1}, t_{n-1}; you don't need to expand all the way.] For the one with functions, you have to go a bit mental with the notation and it's all great fun, but then you can use it to differentiate things like (sin⁡x)(cos⁡x)(tan⁡x)(csc⁡x)(sec⁡x)cot⁡x(\sin x)^{(\cos x)^{(\tan x)^{(\csc x)^{(\sec x)^{\cot x}}}}}.

[I had too much spare time in the summer before uni.]


Any good websites for this sort of thing please? And is it basically coming up with a formula to differentiate things with n powers? :biggrin:
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nuodai
17
Original post by Chemhistorian
Any good websites for this sort of thing please? And is it basically coming up with a formula to differentiate things with n powers? :biggrin:


You don't need anything more than A-level differentiation skills to do it.
Reply 22
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anshul95
11
Original post by nuodai
It's all essentially the same thing. For the first one you pretty much already did it with this one; let y=tn(x)y=t_n(x), and then ln⁡y=tn−1(x)ln⁡x\ln y = t_{n-1}(x)\ln x. [Remember you just need to find a recursive formula in terms of tn,tn−1′,tn−1t_n, t'_{n-1}, t_{n-1}; you don't need to expand all the way.] For the one with functions, you have to go a bit mental with the notation and it's all great fun, but then you can use it to differentiate things like (sin⁡x)(cos⁡x)(tan⁡x)(csc⁡x)(sec⁡x)cot⁡x(\sin x)^{(\cos x)^{(\tan x)^{(\csc x)^{(\sec x)^{\cot x}}}}}.

[I had too much spare time in the summer before uni.]

I just did that question now - I think I changed my system of notation about 6 or 7times before I was happy. I haven't really managed to find a compact way of expressing the end result though.

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