# mathUse implicit differentiation if y= x^2-5xy+3y^2=7

### 2 Answers | Add Yours

To find the derivative dy/dx of y= x^2-5xy+3y^2 implicit differentiation can be used and differentiation carried out in the following way:

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

dy/dx = 2x - 5*x*(dy/dx) - 5y + 6y*(dy/dx)

take all terms with dy/dx to one side

dy/dx + 5*x*(dy/dx) - 6y*(dy/dx) = 2x - 5y

=> dy/dx( 1 + 5x - 6y) = 2x - 5y

=> dy/dx = (2x - 5y)/(1 + 5x - 6y)

**The derivative dy/dx = (2x - 5y)/(1 + 5x - 6y)**

We'll differentiate with respect to x to get dy/dx:

(d/dx)( x^2-5xy+3y^2-7)=0

We'll differentiate each term of the algebraic sum:

(d/dx)( x^2) - (d/dx)5xy + (dy/dx)3y^2 = 0

2x - (5x*(dy/dx) + 5y) + 6y*(dy/dx) = 0

(dy/dx)(5x + 6y) = -2x + 5y

We'll divide by 5x + 6y both sides:

dy/dx = (5y - 2x)/(5x + 6y)