2. Examples of finding the derivative using IMPLICIT DIFFERENTIATION. a. What we have been doing up until now is explicit differentiation, though we typically just say "differentiation" or "finding the derivative." y= x2 +3 = y'=2x, where y' is the first derivative. NOTE: y is written explicitly, meaning that y is directly defined. In other word "y is" an expression with one variable. b. But we can taek the derivative implicitly when y is not written explicitly. 4 = x2 + y2 is defined implicitly, since the equation is not directly stating what is y -7 We could write is explicitly as y= + \\ 4 - x2 but for now we will focus on finding derivative implicitly. 1. Find the derivative implicitly. [ 4 ] = [x2 + v2] dx o = [x2 ] + d [v2 ] Note: when evaluating implicitly, we use our derivative rules as usually, but whenever you take the derivative of y you will follow the result with y' 0 = 2x + 2y- y' Notice the derivative of y2 is 2y multiplied by y'.Note: when evaluating implicitly, we use our derivative rules as usually, but whenever you take the derivative of y you result with y' 0 = 2x + 2y - y' Notice the derivative of V2 is 2y multiplied by y'. 2. Now we will solve for y'. 2x 2y-y' 3. Resulting in the following derivative written implicitly. ,2 2X 2v Y 2. Example of implicit differentiation with the need for the product rule. a. Written example 4x =7x2 +3xy2 1. Find the derivative implicitly. [4x ] = [7x2 + 3xy? ] d [7x2 ] + [3xy? ] dx 4 = 14x+ [3xy2] dx Note: we need to apply the product rule while keeping in mind what we have learned: whenever you take the derivative of y you will follow the result with y' dx [3xy2 ] = 3 d xy? = 3 x dx x d [y ] + y d [x ] 3( x d Ly? ]+vz d [x] =3 x[2y- v'] +v z[1] Back in context: 4= 14x+ -[3xy?] dx 4 = 14x + 3[2xy - y' + y?] 4 =14x +6xy . y' + 3y22. Now we will solve for y'. 4 = 14x + 6xy . y' + 3y2 6xy . y' = 4 - 14x - 3y2 6xy - y' _4- 14x - 3y2 6xy 6xy 3. Resulting in the following derivative written implicitly. y's. 4 - 14x - 3y2 6xyFind the derivatives implicitly. 1. Find 3/ if 2 = x 33/2. 2. Find 3/ if 2x = x2 2y3. 3. Find yr if xzyz 3X4 = 4X2