... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case t), and the straight dashed line (and in this more nested case also the squiggly line) similarly but with respect to the dashed (or squiggly) balloon expression (the inner function of the composite which is subject to the chain rule).

Thank you very much. I think that this teaches me that there doesn't need to be a very "straight-forward" or "systematic" solution to a problem - that we have to invent stuff according to our needs..
Am I correct?

Thank you again for the detailed (and very neat) working.

Oct 12th 2013, 10:38 AM

tom@ballooncalculus

Re: Prove the identity by direct differentiation

Er, thank you! I'll hope that 'neat' means not lacking in 'system' and 'straight-forwardness'. Others may offer a more conventional working, anyhow.