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8. Differentiation of Implicit Functions

by M. Bourne

We meet many equations where y is not expressed explicitly in terms of x only, such as:

You can see several examples of such expressions in the Polar Graphs section.

It is usually difficult, if not impossible, to solve for y so that we can then find .

We need to be able to find derivatives of such expressions to find the rate of change of y as x changes. To do this, we need to know implicit differentiation.

 

Example 1:

Find the expression for if y⁴ + x⁵- 7x² - 5x^-1 = 0.

Answer

Example 2:

Find the slope of the tangent at the point (2,-1) for the curve:

Answer

In LiveMath, we can see the shape of the curve we have been dealing with. We can also see that the slope of the tangent is indeed -4.

LIVEMath

 

Example 3 (Involves Product Rule)

Find the expression for if:

(This is the example given above.)

Answer

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Differentiation
• 1. Limits and Differentiation
• 2. The Slope of a Tangent to a Curve (Numerical)
• 3. The Derivative from First Principles
• 4. Derivative as an Instantaneous Rate of Change
• 5. Derivatives of Polynomials
• 6. Derivatives of Products and Quotients
• 7. Differentiating Powers of a Function
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives

• Comments, Questions?

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