# Implicit Differentiation

Implicit Differentiation examples with detailed solutions are presented

 Example 1: Use implicit differentiation to find the derivative dy / dx where y x + sin y = 1 Solution to Example 1: Use the sum rule of differentiation to the whole term on the left of the given equation. d [xy] / dx + d [siny] / dx = d[1]/dx . Differentiate each term above using product rule to d [xy] / dx and the cain rule to d [siny] / dx. x dy / dx + y + (dy / dx) cos(y) = 0 . Note that in calculating d [siny] / dx, we used the chain rule since y is itself a function of x and sin (y) is a function of a function. Solve for dy/dx to obtain. dy / dx = -y / (x + cos y) Example 2: Use implicit differentiation to find the derivative dy / dx where y 4 + x y 2 + x = 3 Solution to Example 2: Use the differentiation of a sum formula to left side of the given equation. d[y 4] / dx + d[x y 2] / dx + d[x] / dx = d[3] / dx Differentiate each term above using power rule, product rule and chain rule. 4y 3 dy / dx + (1) y 2 + x 2y dy / dx + 1 = 0 Solve for dy/dx. dy/dx = (-1 - y 2) / (4y 3 + 2xy) Example 3: Find all points on the graph of the equation x 2 + y 2 = 4 where the tangent lines are parallel to the line x + y = 2 Solution to Example 3: Rewrite the given line x + y = 2 in slope intercept form: y = -x + 2 and identify the slope as m = -1. The tangent lines are parallel to this line and therefore their slope are equal to -1. The slope of tangent lines at a point can be found by implicity differentiation of x 2 + y 2 = 4 2x + 2y dy/dx = 0 Let P(a , b) be the point of tangency. At point P the slope is -1. Substituting x by a, y by b and dy/dx by -1 in the above equation, we obtain 2a + 2b (-1) = 0 Point P(a , b) is on the graph of x 2 + y 2 = 4, hence a 2 + b 2 = 4 Solve the system of equations: 2a - 2b = 0 and a 2 + b 2 = 4 to obtain two points (-sqrt(2) , -sqrt(2)) and (sqrt(2) , sqrt(2)) Exercises Use implicit differentiation to find dy/dx for each equation given below. 1:     xe y = 3 2:     x 2 + y 2 = 20 3:     x sin(xy) = x solutions to the above exercises 1:     dy/dx = - 1 / x 2:     dy/dx = - x / y 3:     dy/dx = [ 1 - sin(xy) -xy cos(xy) ] / [ x2 cos(xy)] More on differentiation and derivatives