2.11 Implicit Differentiation



Not all curves in the plane (given by an equation in x and y) are the graphs of functions. However, if it is assumed that for a piece of the curve, the equation determines implicitly a differentiable function y = f(x), then it is possible to find the derivative of f without explicitly finding a formula for y in terms of x. The method is called that of Implicit Differentiation.

By the end of your studying, you should know:

On-screen applet instructions: The button at the very bottom gives the interval over which you are tracing where it is possible to define y as a function of x. Use this button only to check your work after you have tried to find the interval on your own.


Find y' by implicit differentiation, where xy = cot(xy).

Find the tangent line to the ellipse

at the point

An interesting curve first studied by Nicomedes around 200 B.C. is the conchoid, which has the equation x2y2 = (x + 1)2 (4 – x2). Use implicit differentiation to find a tangent line to this curve at the point (–1, 0).


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See Exercises for 2.11 Implicit Differentiation (PDF).

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2.10 The Mean Value Theorem Table of Contents 2.12 Derivatives of Exponential and Logarithm Functions

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel