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Directions for Implicit Differentiation

The two fundamental processes in calculus are differentiation and integration. A student has to master differentiation before she can move on to integration. While differentiating, or finding the derivative, you may find that the equation does not have variables expressed explicitly as one in terms of the other. A popular example is the equation for a circle -- (x^2) + (y^2) = 1. For such an equation, you can solve for y in terms of x and then differentiate to get dy/dx, or you can use an easier approach: implicit differentiation. In implicit differentiation, all the terms in the equation are differentiated in terms of one variable (e.g. x), and then the resulting equation can be simplified to find the desired derivative (i.e. dy/dx).

  1. Determine the Desired Derivative

    • Set up the equation and determine the dependent and the independent variable. Then determine the desired derivative form (whether it is dy/dx, dx/dy, or some other form).

    Differentiate Each Term

    • Use the chain rule and any other necessary rules (e.g, power rule, quotient rule, product rule, trigonometric rules for differentiation) to differentiate each term in the entire equation. Find the d/dx of each term under the chain rule. For example, if the equation is (x^2) + (y^2) = 25, do the following: d/dx ( (x^2) + (y^2) = 25 ).

    Simplify

    • Simplifying the aforementioned equation results in d/dx (x^2) + d/dx (y^2) = d/dx (25), which is the same as (2x) + (2y) (dy/dx) = 0.

    Rearrange and Solve for Derivative

    • The final step is to solve for dy/dx by rearranging the terms. So, it would result in (2y) (dy/dx) = -2x, and then dy/dx = -2x/2y = -x/y.

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