Find dy/dx by implicit differentiation. x^2-5xy+3y^2=7.

x^2 - 5xy + 3y^2
= 7

We need to differentiate the equation with respect to x.

==> dy/dx( x^2 - 5xy + 3y^2 ) = dy/dx ( 7)

==> dy/dx (x^2) - 5
dy/dx ( xy) + dy/dx ( 3y^2) = dy/dx ( 7).

We will differentiate with respect
to x.

==> 2x - 5 ( 1*y + xy' ) + 6yy' = 0

==> 2x -
5y - 5xy'+ 6yy' = 0

Now we will combine the terms with y' on the left
side.

==>  6yy' - 5xy' = 5y - 2x

Now we will factor
y'.

==> y'*( 6y - 5x ) = 5y-2x

Now we will divide by
(6y-5x).

==> y' = (5y-2x)/(6y-5x)