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Differentiation Question watch

1. Can someone help me with the following questions please?

1) Given that y = ln (x + √(x² +1)),
show that: dy/dx = 1/ √(x²+1)

2 Find
a) d/dx (ln(lnx)
b) d/dx (sin(sinx))

3) show from first principle that if y = cosx then the diferrential is -sinx

Thanks to anyone that can help me!!
2. For (3) I have a good resource: http://mathscentre.ac.uk/resources/w...principles.pdf
3. 1) Given that y = ln (x + √(x² +1)),,show that: dy/dx = 1/ √(x²+1)
Use the chain rule with u = x + √(x² +1), then y = lnu
a) d/dx (ln(lnx)
Use the chain rule with u = lnx, then y = lnu
b) d/dx (sin(sinx))
Use the chain rule with u = sinx, then y = sinu
3) show from first principle that if y = cosx then the diferrential is -sinx
Check page 4
4. (Original post by latin.snake)
Can someone help me with the following questions please?

1) Given that y = ln (x + √(x² +1)),
show that: dy/dx = 1/ √(x²+1)
That looks like a hyperbolic function...

sinh-1 x = ln (x +√(x² +1))

so y = sinh-1 x
x = sinh y
dx/dy = cosh y
dy/dx = 1/cosh y

cosh y = √(1+sinh² x)

so dy/dx = 1/√(1+sinh² x)

x = sinh y

dy/dx = 1/√(1+x²)

I hope thats right, otherwise I'm failing FP2 tomorrow! It might not be the easiest way to do it but I think its pretty nifty.

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