Home >> Science & Nature >> Science

How to Calculate Implicit Differentiation

In calculus, implicit differentiation addresses mathematical functions where the independent "x" variable doesn't explicitly define the dependent "y" variable --- that is, problems where it's difficult to solve for y in terms of x. Implicit differentiation allows you to find the derivative of such a function without solving the function explicitly for y. One of the rules of differentiation, called the chain rule, must be used when differentiating y. Instruction in the use of the chain rule and other rules of differentiation goes beyond the scope of this article.

Instructions

- 1
  Differentiate both sides of the the equation using the chain rule. Differentiating both sides of the equation y^4 + 3y = 4x^3 + 5x + 1 results in the equation: 4y^3(y') + 3y' = 12x^2 + 5.
- 2
  Manipulate the equation algebraically to isolate the y' terms on one side of the the equation, then simplify. For example, 4y^3(y') + 3y' = 12x^2 + 5 already has y' terms on one side of the equation but can be simplified to: (y')(4y^3 + 3) = 12x^2 + 5.
- 3
  Solve for y' algebraically. For example, solving the equation (y')(4y^3 + 3) = 12x^2 + 5 for y' finds: y' = (12x^2 + 5) / (4y^3 + 3).
- 4
  Substitute the x and y values of a coordinate point into the equation to determine the slope of the function at that point. For example, to find the slope of the point (3, 8) for the function f(x) = y^4 + 3y = 4x^3 + 5x + 1 with derivative f '(x) = y' = (12x^2 + 5) / (4y^3 + 3), substitute x and y into the equation: y' = 12(3)^2 + 5 / 4(8) + 3) = 108 + 5 / 32 + 3 = 113 / 35 = 3.2.

Science