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# Substitution and second order differentials watch

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1. I'm having trouble with part of a FP2 question.

z=sin x implies (dz/dx)=cosx

(dy/dx)= (dy/dz) * cosx

And
(d²y/dx²)= (d²y/dz²) * cos²x – (dy/dz) * sinx

How has the answer to (d²y/dx²) been figured out? Any help would be greatly appreciated. Thanks

PS: It's the "cos²x" part I'm struggling to understand.
2. dy/dx= dy/dz dz/dx chain rule
d2y/dx2= d/dx(dy/dx)= d/dx (dy/dz dz/dx) by definition

use product rule to get

d2y/dx2= dy/dz d2z/dx2 + d/dx(dy/dz) dz/dx
= dy/dz d2z/dx2 + d/dz(dy/dz) dz/dx )dz/dx by chain rule

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Updated: November 28, 2011
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