(I might be wrong on this historical note). The famous mathematician, G.H. Hardy wrote in the book A course in Pure Mathematics a proof of the chain rule.
If are differenciable at . Then, . He made the most simple proof.
What he meant was this....
The first fraction is the derivative of only evaluated at . The second fraction is derivative of evaluated at .
But in the next edition of the book he omitted the proof.
Why? What ailed him?