Answer to Question #186273 in Calculus for noman

Question

Use the limit definition to find partial derivative of

 f(u,v,w)=u^2-nuv+(n+1) v^2-(n+2)uw+(n+3)vw^2-(n+4)u+(n+5)v-(n+6)w,Where n is the sum of your arid number e.g if 20-arid-470 choose n=4+7+0=11. 

Answer 1

 $f(u,v,w)=u^2-nuv+(n+1)v^2-(n+2)uw+(n+3)vw^2-(n+4)u+(n+5)v-(n+6)w$

$f_u=lim_{h\to 0} \dfrac{f(u,h,v,w)-f(u,v,w)}{h}$

 $f_u=lim_{h\to 0}\dfrac{ (u+h)^2-n(u+h)v+(n+1)v^2-(n+2)(u+h)w+(n+3)vw^2-(n+4)(u+h)+(n+5)v-(n+6)w-f(u,v,w)}{h}$

   $=lim_{h\to 0}\dfrac{(u+h)^2-u^2-nv(u+h-u)+(n+2)w(u+h-u)-(n+4)(u+h-4)}{h}$

$=lim_{h\to 0} \dfrac{h((2u+h)-nv+(n+2)w-(n+4))}{h}$

   $=lim _{h\to 0} 2u+h)-nv+(n+2)w-(n+4)$

  

   $=2u-nv+(n+2)w-(n+4)$

Similarly we get $f_v=-nu+(n+1)2v+(n+3)w^2+(n+s)$

      and $f_w=-(n+2)v+(n+3)v(2w)-(n+6)$