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
• Derivative interactive graph
• Differentiation Java applet
• Derivative of a Cubic - Exploration using GeoGebra
• 7. Differentiating Powers of a Function
• 8. Differentiation of Implicit Functions
• 9. Higher Derivatives
• 10. Partial Derivatives
• Differentiation Problem Solver

• Comments, Questions?

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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:

y⁴ + 2x²y² + 6x² = 7

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 `(dy)/(dx)`.

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.

Let's learn how this works in some examples.

Example 1

Find the expression for `(dy)/(dx)` if y⁴ + x⁵ − 7x² − 5x^-1 = 0.

Answer

Example 2

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

2y + 5 − x² − y³ = 0.

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 `(dy)/(dx)` if:

y⁴ + 2x²y² + 6x² = 7

(This is the example given at the top of this page.)

Answer