Issue with implicit differentiation

[cole92]

***EDIT*** just after posting that I can alreay tell my first line is incorrect, but I am still lost as to what I need to do....
I don't think I have properly learned how to do implicit differentiations. I have been working on my homework, and although I get close to the right answer sometimes, I have yet to get onecorrect. Can anyone walk me through a problem please? The trig ones are especially killer.

My work is in red (black is original problem), but I think I am way off the mark.

 

[BigGlenntheHeavy]

\(\displaystyle [sin(\pi x)+cos(\pi y)]^{2} \ = \ 2\)

\(\displaystyle 2[sin(\pi x)+cos(\pi y)]^{1}[\pi cos(\pi x)-y \ ' \ \pi sin(\pi y)] \ = \ 0\)

\(\displaystyle [2sin(\pi x)+2cos(\pi y)][\pi cos(\pi x)-y \ ' \ \pi sin(\pi y)] \ = \ 0\)

\(\displaystyle 2\pi sin(\pi x)cos(\pi x)+2\pi cos(\pi x)cos(\pi y)-2\pi y \ ' \ sin(\pi x)sin(\pi y)-2\pi y \ ' \ sin(\pi y)cos(\pi y) \ = \ 0\)

\(\displaystyle 2\pi y \ ' \ sin(\pi x)sin(\pi y)+2\pi y \ ' \ sin(\pi y)cos(\pi y) \ = \ 2\pi sin(\pi x)cos(\pi x)+2\pi cos(\pi x)cos(\pi y)\)

\(\displaystyle y \ ' \ = \ \frac{2\pi sin(\pi x)cos(\pi x)+2\pi cos(\pi x)cos(\pi y)}{2\pi sin(\pi x)sin(\pi y)+2\pi sin(\pi y)cos(\pi y)}\)

\(\displaystyle y \ ' \ = \ \frac{cos(\pi x)[2 \pi sin(\pi x)+2\pi cos(\pi y)]}{sin(\pi y)[2\pi sin(\pi x)+2 \pi cos(\pi y)]} \ = \ \frac{cos(\pi x)}{sin(\pi y)}\)


If you have a Maple program, you can avoid all the grunt work, viz.

f:=(sin(Pi*x)+cos(Pi*y))^2=2;

> implicitdiff(f,y,x);

cos(Pi x)
---------
sin(Pi y)

 

[cole92]

I'm a little confused as to what happened between your second and third step..

 

[BigGlenntheHeavy]

(a+b)(c+d) = ac+ad+bc+bd

 

[cole92]

Oh okay. I wasn't sure if you used "FOIL" or not, but you did. Thank you. I'll use this as a guide and will hopefully be able to complete some other problems now.

Thanks again.