Expert: NUR ARINA BASILAH BINTI KAMISAN - 7/21/2008

Question
"QUESTION: I had asked this question originally on the site:www.thestudentroom.co.uk But,I couldnot understand the answer as they said to apply chain rule of differentiation.Since the question is not like the general question on which we apply chain rule,I couldnot understand it.The attached picture on this question can be found on this link:http://www.thestudentroom.co.uk/attachment.php?attachmentid=56069&d=1216384477


The question goes like this-
In triangle PQR,
(chord PQ)^2=(PR)^2+(QR)^2=(delta x)^2+(delta y)^2
But,
how come this
(chord PQ/delta s(arc AP))^2*(delta s/delta x)^2=1+(delta y/delta x)^2
Please help me through this since I am facing lots of problem on this.

The picture can be found on this site:
http://img99.imageshack.us/img99/5660/cartician20formrp5.gif"


Answer
dear roshan,

i could not open the link on the studentroom website because they asked for log on id and i don't have one.so here is the answer.

you have a triangle.just imagine the triangle PQR ok.because PQR is a right angle triangle,we can apply the theorem pythagoras.(for more info about theorem pythagoras,u can visit this website http://en.wikipedia.org/wiki/Pythagorean_Theorem)

from theorem pythagoras the hypotenuse will be the chord PQ and the other two side will be QR and PR.so,when we substitute in the theorem we will get:

(chord PQ)^2=(PR)^2 + (QR)^2 which was also equal to
           =(delta x)^2 + (delta y)^2

now,chain rule is something like this:

dy/dx=(dy/du) x (du/dx)
u can cancel out the du and will get dy/dx (try to write it back in a paper.so u can see it more clear).now back to your question,you have an equation like this:

(chord PQ/delta s (arc AP))^2*(delta s/delta x)^2
=(chord PQ)^2/(delta s(arcAP))^2 * (delta s)^2/(delta x)^2
(note that delta s(arc AP)=delta s)
=(chord PQ)^2/(delta s)^2 * (delta s)^2/(delta x)^2 -->(here where we apply the chain rule,we can cancel out the delta s^2)
=(chord PQ)^2/(delta x)^2
and because (chord PQ)^2=(delta x)^2 + (delta y)^2,substitute it for (chord PQ)^2 in the equation above and u will have:
=[(delta x)^2 + (delta y)^2]/(delta x)^2
= (delta x)^2/(delta x)^2 + (delta y)^2/(delta x)^2
= 1 + (delta y)^2/(delta x)^2
= 1 + (delta y/delta x)^2 #proved

-->try to write it back in a paper and you can understand it