Expert: Ahmed Salami - 10/20/2004

Question
This is the problem: xcosy=y. I am suppose to find d^2y/dx^2 of xcosy=y. I was able to find dy/dx of xcosy=y but I am having trouble finding the derivative of that derivative. Thanks in advance for the help.


Answer
Hi Nick,
I'm going to start from the beginning. All we do is find the derivative of every expression with respect to x. Note that when we do this for any expression in y we simply add dy/dx.
Now,
xcosy = y
Differentiating both sides gives
x(-siny)dy/dx + cosy = dy/dx
cosy = dy/dx + xsiny dy/dx
cosy = dy/dx(1 + xsiny)
where we get dy/dx = cosy/1 + xsiny
but continuing with the differentiation without isolating y, we get
-siny dy/dx = dy/dx[xcosy dy/dx + siny]
             + (1 + xsiny)d^2y/dx^2
(1 + xsiny)d^2y/dx^2 = -siny dy/dx - dy/dx[xcosy dy/dx + siny]
(1 + xsiny)d^2y/dx^2 = -dy/dx[xcosy dy/dx + 2siny]
(1 + xsiny)d^2y/dx^2 = -[xcosy(dy/dx)^2 + 2siny dy/dx]
d^2y/dx^2 = -[xcosy(dy/dx)^2 + 2siny dy/dx]/(1 + xsiny)
Now you just insert the expression for dy/dx and simplify for the complete answer.
I hope you get it. Let me know if there's still anything unclear. Any delay regretted.
Regards.