# Thread: Prove the identity by direct differentiation

1. ## Prove the identity by direct differentiation

Hi, please have a look at the attached question and the attempt made to solve, but I can't
The question is Problem 1.2 (b)

The vectors in my working are underlined, but are bold in the question.

The thing is, that I don't know where the c springs from in the RHS.
Here is the question:

My attempt at solving:

Thanks for the help

2. ## Re: Prove the identity by direct differentiation

Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

... 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).

... is the product rule, where, again, straight continuous lines are differentiating downwards with respect to t.

Full size

__________________________________________________ __________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

3. ## Re: Prove the identity by direct differentiation

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.

4. ## 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.

Cheers

Tom