Write an expression for the slope of the curve of x^2 - 2xy +4y^2 = 64 at any point

help me please, i dont remember how to do this and its for a grade, please show me how to do this.

alright, so this equation has to be differentiated implicitly. thats just a fancy way of saying it has both x and y terms in it so for every y term you have to treat it as its own function (ie. the deriv of y isnt just 1, its y').

so, start with the first term x^2, who's derivative is 2x

now the second term should be evaluated with the product rule: it will be -2 (xy' + y)

now the third term whos derivative its 8yy'. this is beacuse you take the deriv of the outside, 8y, times the inside, y'.

lastly we take the deriv of the right side, 64, which is 0 because the deriv of any constant is 0

now stringing these all together you get 2x-2(xy'+y)+8yy'=0

simplifying this you get 2x-2xy'-2y+8yy'=0

grouping the y' terms together and factoring it out you get y'(-2x+8y)+2x-2y=0

move all the other terms to the right and divide by the term adjacent to y' and you will end up with (2y-2x)/(-2x+8y)

simplify by factoring a 2 out of the top and bottom and you end up with your final answer: y'= (y-x)/(4y-x)

hope i helped!!