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6. Derivatives of Products and Quotients

by M. Bourne

PRODUCT RULE

If u and v are two functions of x, then the derivative of the product uv is given by...

In words, this can be remembered as:

"The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first."

Example 1:

If we have a product like

y = (2x² + 6x)(2x³ + 5x²)

we can find the derivative without multiplying out the expression on the right.

Answer

LiveMath does things in a slightly different order...

LIVEMath

 

Example 2:

Find the derivative of

Answer

NOTE:

We can write the product rule in many different ways:

OR

… etc.

 

QUOTIENT RULE

(A quotient is just a fraction.)

If u and v are two functions of x, then the derivative of the quotient u/v is given by...

In words, this can be remembered as:

"The derivative of a quotient equals bottom times derivative of top minus top times derivative of the bottom, divided by bottom squared."

Example 1:

We wish to find the derivative of the expression:

Answer

Play with this example on the Differentiation Java applet page.

Example 2:

Find if .

Answer

Play with this example on the Differentiation Java applet page.

Let's also see this in LiveMath.

LIVEMath

NOTE:

You will also see the quotient rule written as:

OR

etc.

 

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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