There’s some more advanced differentiation. This comes in two parts, the first of which depends on knowing some integrations. That feels odd, because we began by learning that differentiation can always be done. There are, however, some functions whose integral is recognised first. One such would be ∫1/x dx -> ln x.
I’ve broken the topic into parts:
(i) A page on the hyperbolic functions
(ii) The differentiation/integration of hyperbolics
(iii) The extended use of differentiation
So here (hard to type) are some standard integrals that demonstrate skills in substitution and are all worth learning for C4 perhaps but especially the equivalents under other Boards.
1. (1+x2)-1 Substitute tan u = x. Use 1+ tan2u = sec2u and sec2u du = dx
2. (1-x2)-1/2 Substitute sin u = x. Follow the pattern set already.
3. -(1+x2)-1/2 Substitute cos u = x.
4. (a2-x2) -1
The next four follow on from the previous four:
5. Use the knowledge gained from Q1 to integrate (a2-x2)-1/2
6. Write down the integral of -(1+x2)-1/2
7. Write down the integral of (a2+x2)-1 . Check your solution by doing the substitution.
8. (x2-a2) -1
Q3 follows from Q2 but Q5 begs the question what is the integral of (x2-a2)-1/2 . The square root around the negative of what has gone before suggest that a multiplier of I [i2=-1] will be required.
If you’re stuck on the use of substitution then (i) you probably shouldn’t be reading this page and (ii) you need to sort that out promptly (immediately, now) perhaps by reading this page here.
In terms of Edexel modules, the hyperbolic functions are within FP3; the textbook devotes Chapters 1 & 3 to that, 15% of the book and at least that much of the relevant exam paper. The well-prepared student examines what is the same and what is different and picks up those similarities and differences so that ‘hyperbolics’ fit into the total known.
Much more advanced differentiation.
The CIE syllabus picks out some tasks in differentiation rarely seen at school.
Related pages (on their way)
Hyperbolics (next)
Adv Integration
CIE differentiation