Implicitly differentiate the following equation to find dy/dx in terms of x and y: 2x^2y + 2x + 4y – cos (piy) = 17

Firstly remember that each part of the equation can be differentiated separately. Let's label each part:

Part A: 2(x^2)y

Part B: 2x

Part C: 4y

Part D: -cos(piy)

Part E: 17

Parts B and E are easy to differentiate, and go to 2 and 0 respectively. For part C we have to implicitly differentiate it, so the y goes to dy/dx, leaving us with 4(dy/dx). Part A is slightly trickier, since the single term contains both x and y. For this, we have to use the product rule (which is u'v + uv'). 2x^2 differentiated is 4x, and y differentiated is dy/dx. This leaves us with 4x X y + 2x^2 X dy/dx, or 4xy + 2x^2(dy/dx). Finally we get onto part D, where we have to use the chain rule (remember that it's negative cos!). -cos goes to +sin, then we differentiate whatever's in the brackets of the cos and multiply the sin by that. dy/dx of piy is pi(dy/dx), so we're left with pi(sin(piy))(dy/dx).

Overall the implicitly differentiatied equation is 4xy + 2x^2(dy/dx) + 2 + 4(dy/dx) + pi(sin(piy))(dy/dx) = 0

But we're not done yet! The question asks for dy/dx in terms of x and y, so we have to rearrange the equation. First put all the dy/dx terms on one side, then factor out the dy/dx.

(dy/dx)(2x^2 + 4 + pi(sin(piy))) = - 4xy - 2

Then simply divide the left hand side by (2x^2 + 4 + pi(sin(piy))) to get the equation in terms of dy/dx:

dy/dx = (- 4xy - 2)/(2x^2 + 4 + pi(sin(piy)))