Calculus I using LiveMath

Chapter 3

The Quotient Rule

You just found in the last section that the derivative of the product of two functions is not the product of their derivatives. Likewise, the derivative of the quotient of two functions is not the quotient of their derivatives.

LiveMath, again, makes it easy to demonstrate a rule of differentiation. As with the past couple of examples, you will use the substitute variables u and v to do this.

Example 3.9 The Quotient Rule

• Input the following two Props. The first defining a function of a quotient; the second, the Derivative Op. Remember, the derivative is taken with respect to x. You are assuming that both u and v are functions of x. You put the function in this form so you will get the intermediate answer showing you the rule.
  
• Substitute the first Prop into the second (with Auto-Simplify on).
  

To see this in action, define u and v as simple functions of x.

• Try the following two.
  
• Substitute the first (u = x^3) and second (v=5x) Props above into the Quotient Rule you achieved in the step above. For the first step make sure Auto-Simplify is turned off. This will show the substitution before it dissolves into the answer. Use an Expand for the final answer.
  
• You can have LiveMath perform the differentiation in one step by making the y-function contain the smaller functions from the start.
  

Click on the LiveMath Internet icon below to see this last example in action.

LiveMath Internet - Quotient Rule Demo

The next example uses more complicated expressions to demonstrate the Quotient Rule.

Example 3.10 The Quotient Rule using real functions

• Differentiate the following function in a way that demonstrates the Quotient Rule.
  

The subject of how to differentiate e^x is covered later in the book. So, at this point, let LiveMath do the work and observe the results. Careful observation may peak your curiosity in this example.

• Set up the example as you did above making sure that, while you perform the manipulations, you remember to switch Auto-Simplify on and off at the appropriate times.
  
• The final answer is shown below. The intermediate manipulations are left to you. You will find the Expand manipulation useful in this example, along with a Collect, which produces the final answer shown below.
  

What would you do if the numerator of the quotient was equal to 1? In other words, how would you differentiate the following function?

To find out, and to see how it extends the scope of the power rule, click here.