The curve C has the equation 4x^2 - y^3 - 4xy + 2y = 0 . The point P with coordinates (-2, 4) lies on C. Find the exact value of dy/dx at the point P.

4x² - y³ - 4xy + 2^y = 0To find dy/dx we need to differentiate all the terms in the equation. As you may notice the x's and y's are mixed together so we will have to use implicit differentiation. Since we are differentiating in terms of x, all of the x terms can be differentiated as usual, whilst the y terms will be followed by a dy/dx. The first two terms are more straight forward to deal with: d(4x²)/dx = 2 * 4 * x = 8xd(y³) / dx = 3 * y²* dy/dx = 3y²dy/dx[Term 3] For the composite term, 4xy, we will be using the product function rule: y = f(x)g(x), y' = f'(x)g(x) + f(x)g'(x). So: d(4xy)/dx = 41y + 4x1dy/dx = 4y + 4x dy/dx.[Term 4] In order to differentiate the 2^y, we will be using the general rule for differentiating exponents which is y = a^f(x), y' = a^f(x)* f'(x) ln(a). So : d(2^y)/ dx = 2^y 1 dy/dx * ln2 = 2^y ln2 *dy/dx .Combining all of our work from above:8x - (3y²dy/dx ) - ( 4y + 4xdy/dx) + ln2(2^y*dy/dx) = 08x - 3y²dy/dx - 4y - 4xdy/dx + ln2(2^y*dy/dx) = 0 Now we have the differentiated equation so we can just plug in the coordinate values into x and y! 8(-2) - 3(4)²*dy/dx - 4(4) - 4(-2)*dy/dx + ln2(2⁴*dy/dx) = 0 Rearranging everything in terms of dy/dx:dy/ dx = - 32 / (40 - 16ln2)