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) ] / [ x^{2} cos(xy)]
More on differentiation and derivatives 
