Find dy/dx with implicit differentiation (xy)^1/2=1+x^2y

Question

ranga · Answer

use power rule, chain rule and product rule.

anonymous · Answer

Can you show me?

amistre64 · Answer

let a = xy
    a' = x'y + xy'  by product rule

let b = x^2y ... which is either menas x^2 y or x^(2y)  hard to tell
     b' = derive it as needed

then assess:  a^(1/2) = 1 + b
        1/2 a^(-1/2) a' = 1 + b'

anonymous · Answer

It's x^2 y

amistre64 · Answer

then thats pretty much a product rule as well ... how would you define b' with that?

anonymous · Answer

2xy+x^2 dy/dx ?

amistre64 · Answer

correct .. but i like the ' notations

b' = 2x x' y + x^2 y'

the reason i keep the x' parts is to show the chain rule in action.   now x' = dx/dx = 1 to clean it up afterwards

amistre64 · Answer

let a = xy
    a' = x'y + xy'  by product rule

let b = x^2 y
     b' = 2x x' y + x^2 y'

             a^(1/2) = 1 + b
 1/2 a^(-1/2) a' = 1 + b' 

the ab was just for a cleanup ... to easier see an overall picture, sub back in the information

 1/2 (xy)^(-1/2)  (y+xy') = 1 + (2xy + x^2 y') 

the rest is just algebra to solve for y'

anonymous · Answer

Okay I get it now thank you very much :)