Implicit Differentiation Implicit and Explicit functions An explicit function is an function expressed as y = f(x) such as         y  =  2x3 + 5 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 (2,2) if          xy + x/y  =  5 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  =  2 and y  =  2,         2y' + 2 + (2 - 2y')/4  =  0         Multiply both sides by 4         8y' + 8 + 2 - 2y'  =  0         6y'  =  -10         y'  =  -5/3 Exercises: Let             3x2 - y3   =   4x + y2 Find dy/dx Find dy/dx at (-1,1) if         x + y  =  x3 + y3   Find dy/dx if         x2 + 3xy + y2 = 1 Find y'' if         x2 - y2  =  4     Application   Example Suppose that the demand function for a boat shop is given by          p  =  -0.01x3 + x + 10,000Find the rate of change of x with respect to p when x  =  20.  A boat craftsman can think of this question as who fast will the number of boats she will need to build change as the price is increased.  Solving for x in terms of p is nearly impossible.  Instead, we can differentiate implicitly.        1  =  -0.03x2 x' + x'Now plug in 20 for x to get        1  =  -0.03(20)2 x' + x'        1  =  -12x' + x'  =  -11x'        x'  =  -1/11The rate of change is -1/11 boats per dollar increase.   e-mail Questions and Suggestions