Differential calculus problem

I am studying calculus at home as a hobby. I have 2 books: The Complete Idiot's Guide to Calculus by W. M. Kelley (K), and Calculus Made Easy (1998)by S.P.Thomson and M.Gardner (T).

On page 119 of (K) a formula is derived which seems to show that the derivative of the inverse function equals the reciprocal of the derivative of the inverse function! This is contradicted by (T) page 148: "...the reciprocal of the derivative of the inverse function gives the derivative of the function itself. The latter version seems to me to be correct, but I cannot see where the other version goes wrong when it is produced in the "Critical Point " box on p119 of (K).

It may be that I am missing the point because of the 2 different notations which are being used. Unfortunately I cannot access on my webpage any maths symbols, hence the description above. I would be grateful for an explanation of my problem and also some advice as to how I access maths symbols.

Differential Calculus problem

Dear Folks

Thanks for answers, which have made the maths clear to me.

However, I still have a small problem with the dash notation, but I cannot spell this out without acess to a maths symbol list. How do you write maths on this website - you all clearly have access to a symbol list?

Biggles (maths tyro)

Differential Calculus problem

$\displaystyle Is (f^-1)' (x) the same as f'(f^-1(x)) $ ?

Differential Calculus problem

Dear Folks

After reading all the posted replies, and worrying for days over the problem of the apparent conflict between the 2 different versions of the equation, I confess that I am totally baffled!

$\displaystyle \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$ seems very clear, and is equal to: " The derivative of the function is equal to the reciprocal of the derivative of the inverse function."

If that is all correct, then I cannot understand the alternative version in the dash notation: $\displaystyle (f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}$.

This version seems to say: "The derivative of the inverse function is equal to the reciprocal of the derivative of the inverse function", which seems to contradict the first version, and indeed contradict itself!

I can see how both versions are derived, but have been unable to sort out the apparent contradiction, which must I think be due to an error in my understanding of the dash notation. I think I must be misunderstanding the meaning of the left hand side of the dash equation.

Please bear with my inability to resolve this problem.

Biggles