The curve C has equation: 2x^2y + 2x + 4y – cos (piy) = 17. Use implicit differentiation to find dy/dx in terms of x and y.
You want to take the derivative of both sides of the equation with respect to x which produces an equation in terms of x, y and dy/dx which can then be rearanged. The right side is just 0 since 17 is a constant. Now look at the first term on the left (2x2y). This requires the product rule to differentiate as follows: d/dx(2x2).y + d/dx(y).2x2 which results in 4x . y + 2x2.dy/dx. The next two terms are simple and so 2x -> 2 and 4y -> 4dy/dx. The final term should be differentiated as would normally be done for cosine giving - - d/dx(piy).sin(piy) which equals +pidy/dx.sin(piy) being careful with the signs.Now we have this equation: 4xy + 2x2dy/dx + 2 + 4dy/dx + pi.dy/dx.sin(piy) = 0 and we need to rearrange it to have just dy/dx on one side. Move all terms without dy/dx onto the right and then factorise dy/dx out of the remaining terms to get: dy/dx(2x2 + 4 + pi.sin(piy)) = - 4xy - 2. Then divide both sides of the equation by the term in brackets to get the answer.dy/dx = (-4xy -2)/(2x2+ 4 + pi.sin(piy))
