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?

Summer holidays

Recommendation

Easy to understand calculus lessons on DVD. Try before you commit. More info:
MathTutorDVD.com

Online Algebra Solver

Solve your algebra problem step by step!

Online Algebra Solver »

Share

Share this page.

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...

`(d(uv))/(dx)=u(dv)/(dx)+v(du)/dx`

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."

Where does this formula come from? Like all the differentiation formulas we meet, it is based on derivative from first principles.

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

Example 2

Find the derivative of

y = (x³ − 6x)(2 − 4x³)

Answer

NOTE:

We can write the product rule in many different ways:

`(d(uv))/(dx)=uv’+vu’`

OR

`(d(fg))/(dx)=f(x)d/(dx)g(x)+g(x)d/(dx)f(x)`… 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...

`d/(dx)(u/v)=(v(du)/(dx)-u(dv)/(dx))/(v^2`

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 3

We wish to find the derivative of the expression:

`y=(2x^3)/(4-x)`

Answer

Play with this example on the Differentiation Java applet page.

Example 4

Find `(dy)/(dx)` if `y=(4x^2)/(x^3+3)`

Answer

Interactives

You can explore Example 4 to see what we've found in:

• Derivatives interactive graph (also works on tablets, like iPad)
• The Differentiation Java applet page. (non-mobile)

Other ways of Writing Quotient Rule

You can also write quotient rule as:

`d/(dx)(f/g)=(g\ (df)/(dx)-f\ (dg)/(dx))/(g^2`

OR

`d/(dx)(u/v)=(vu'-uv')/(v^2)`