Implicit Differentiation Implicit and Explicit Functions An explicit function is an function expressed as y = f(x) such as         y = sinx y is defined implicitly if both x and y occur on the same side of the equation such as         x2 + y2 = 4 we can think of y as function of x and write:         x2 + y(x)2 = 4 To find dy/dx, we proceed as follows: Take d/dx of both sides of the equation remembering to multiply by y' each time you see a y term. Solve for y' Example Find dy/dx implicitly for the circle          x2 + y2 = 4 Solution         d/dx (x2 + y2)  =  d/dx (4) or         2x + 2yy'  =  0 Solving for y, we get         2yy'  =  -2x         y'  =  -2x/2y         y'  =  -x/y Example:   Find y' at (4,2) if          xy + x/y  =  10 Solution:           (xy)' + (x/y)' = (5)' Using the product rule and the quotient rule we have                         y - xy'         xy' + y +                =  0                             y2 Now plugging in x = 4 and y = 2,                          2 - 4y'         4y' + 2 +                =  0                             22             16y' + 8 + 2 - 4y' = 0         Multiply both sides by 4         12y' + 10  =  0         12y' = -10         y' = -5/6   Exercises: Let             3x2 - y3  = 4x cosx + y2 Find dy/dx Find dy/dx at (-1,1) if         (x + y)3 = x3 + y3   Find dy/dx if         x2 + 3xy + y2 = 1 Find y'' if         x2 - y2 = 4 Back to Math 105 Home Page