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)]


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