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Differentiate 5x^2+5y^2-6xy=13 to find dy/dx

This is an example of implicit differentiation, where the function in question involves two variables (x and y) and it is hard to rearrange to create a neat "y=..." equation. To differentiate this equation we must differentiate with respect to x (w.r.t.x). The first term of the equation is simple to differentiate, 5x^2 becomes 10x. However the second term 5y^2 is trickier, in this case we keep the 5 constant but the derivative of y^2 w.r.t.x is 2y dy/dx. This is because the derivative of y^2 w.r.t.y is 2y, which we then multiply by dy/dx. This term is therefore 5 times 2y dy/dx so 10y dy/dx.The third term we use the product rule to find the derivative of -6xy is -6x dy/dx - 6y (the basic derivative of y is dy/dx).The 13 on the other side of the = sign is differentiated to 0 as it is only a constant. We are now left with 10x + 10y dy/dx -6x dy/dx -6y =0. Rearranging this for dy/dx we get the answer for dy/dx as (6y-10x)/(10y-6x), or simplified further (3y-10x)/(5y-3x)

Answered by Henry S. Maths tutor

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