How can I find the derivative of tan(5/x) ?

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How can I find the derivative of tan(5/x) ?

Mathematics
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if you know the derivative of tan u and the derivative of 5/x, you can use the chain rule.
by letting u = 5/x
I don't know the derivative of tan u xD Is there any way to find it without having it memorized?

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no need for u
it's a really good one to know, especially for integration
\(\large\color{black}{ \displaystyle \frac{d}{dx}\left(~\tan(x/5)~\right) }\)
how would you differentiate if it was just tan(x) ?
I don't know lol I get confused when it comes to tan
what is the derivative of tan(x), you don't know ?
Nope xD
you can use a quotient rule, do you know what a quotient rule is?
@Tracy96 cheers
Do you know what d/dx means ?
Ohhh now I remember it was sec^2x hahaha
Working on math too much today XD lol
yes, derivative of tan(x) is sec^2(x)
And when you have tan(x/5) the only difference is that it is sec^2( x/5 ), AND you need to multiply that times the chain rule..... (chain rule for x/5)
For example, \(\large\color{black}{ \displaystyle \frac{d}{dx}\left(~\tan (4x)~\right)=\sec^2(4x)\times \left(\frac{d}{dx}~4x\right) }\) \(\large\color{black}{ \displaystyle \frac{d}{dx}\left(~\tan (4x)~\right)=\sec^2(4x)\times ~4 }\) \(\large\color{black}{ \displaystyle \frac{d}{dx}\left(~\tan (4x)~\right)=4\sec^2(4x) }\)
Ok so that will be \[\sec^2(5/x)(\frac{ -5 }{ x^2 })\]
yes, that is correct, but I would put the last part in front to avoid any confusion
confusion*
Right! Thanks so much @SolomonZelman :D
Sure. Any questions you have regarding this problem or any of the rules?
Nope! Thanks! That was very helpful :)
Alright. You are always welcome!
:)

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