Find the exact value of dy/dx at (-2,4) of the curve C: 4x^2 -y^2 + 6xy + 2^y = 0

First notice that this is an equation that will require implicit differentiation since C cannot be explicitly written in terms of either x or y. Thus we must differentiate each term with respect to x:-the first term is easy as it is in terms of x only, so d(4x^2)/dx = 8x-the second term isn't too hard as it is terms of y only, so d(-y^2)/dx = d(-y^2)/dydy/dx [by chain rule] = -2ydy/dx-the third term is tricker as it is a product of x and y, so d(6xy)/dx = 6xd(y)/dx+6d(x)/dxy [by product rule] = 6xdy/dx + 61y = 6xdy/dx + 6y-the fourth term isn't too hard again as it is in terms of y, but you need to be familiar with standard differentiation results, so d(2^y)/dx = d(2^y)/dydy/dx = (2^y)ln2dy/dx [by standard result]So the final answer for C differentiated with respect to x is:8x - 2ydy/dx + 6xdy/dx + 6y + (2^y)ln2dy/dx = 0Substituting the value of (x,y) = (-2,4) gives:8(-2) - 2(4)dy/dx + 6(-2)dy/dx + 6(4) + (2^4)ln2dy/dx = 0-16 -8dy/dx -12dy/dx +24 +16ln2dy/dx = 08 - 20dy/dx + 16ln2dy/dx = 02 - 5dy/dx + 4ln2dy/dx = 0Now factorising and rearraging gives: dy/dx = 2/(5-4ln2)