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# p5 differentiation watch

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1. how do i differentiate arsechx, arcothx and arcosechx

cheers.
2. y = arsechx
sechy = x

Now implicit differentiation and see how you get on from there...

(same for the others)
3. yeah ive done that much but I get 1/-tanhxssechx for the first one, what do I do with that?
4. i mean 1/-tanhysechy
5. (Original post by Bebop)
i mean 1/-tanhysechy
Use the identity (tanhy)^2+1=(sechy)^2 to rewrite it in terms of sechy and hence x.
6. And at the same time your learning which substitution to use to integrate the ugly looking result!
7. ah right so for the first one its 1/x(x^2 - 1)^1/2 yah?
8. (Original post by Bebop)
how do i differentiate arsechx, arcothx and arcosechx

cheers.
y=sech^-1 x
sechy =x
-sechytanhydy/dx=1

dy/dx=-1/sechytanhy...............(1)
cosh^2 y -sinh^2 y=1
=> 1-sech^2y=tanh^2y
(1) becomes
dy/dx=-1/x(1-x^2)^1/2
ii)
y=cosech^-1 x
cosechy =x
-cosechycothydy/dx=1

dy/dx=-1/cosechycothy...............(2)
cosh^2 y -sinh^2 y=1
=> 1+cosech^2y=coth^2y
(2) becomes
dy/dx=-1/x(1+x^2)^1/2

iii)
y=coth^-1 x
cothy =x
-cosech^2ydy/dx=1

dy/dx=-1/cosech^2y...............(2)
cosh^2 y -sinh^2 y=1
=> coth^2y-1=cosech^2y
(2) becomes
dy/dx=-1/(x^2-1)=1/(1-x^2)

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