Question

Find the derivative of the implicit function: `y^2 - 2xy +sin (x+y) = cos a`?

Answer 1

`(dy)/(dx)=(2y-cos(x+y))/(2y-2x+cos(x+y))`

Explanation:

Implicit differentiation is a special case of the chain rule for derivatives. Generally differentiation problems involve functions i.e. `y=f(x)` - written explicitly as functions of `x`.

However, some functions y are written implicitly as functions of `x`. So what we do is to treat `y` as `y=y(x)` and use chain rule. This means differentiating `y` w.r.t. `y`, but as we have to derive w.r.t. `x`, as per chain rule, we multiply it by `(dy)/(dx)`.

Hence derivative of `y^2-2xy+sin(x+y)=cosa` is

`2y(dy)/(dx)-2(1*y+x*(dy)/(dx))+cos(x+y)(1+(dy)/(dx))=0`

`2y(dy)/(dx)-2y-2x(dy)/(dx)+cos(x+y)+(dy)/(dx)cos(x+y)=0`

i.e. `(dy)/(dx)(2y-2x+cos(x+y))=2y-cos(x+y)`

and `(dy)/(dx)=(2y-cos(x+y))/(2y-2x+cos(x+y))`