The Calculation Thought Experiment (CTE)


To deal with complicated mathematical expressions, we use the following little secret desribed in Calculus Applied to the Real World and Finite Mathematics and Calculus Applied to the Real World, called the Calculation Thought Experiment:

Calculation Thought Experiment

The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, or difference. Given such an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient, and so on.

Using the Calculation Thought Experiment (CTE) to Differentiate a Function
If the CTE says, for instance, that the expression is a sum of two smaller expressions, then apply the rule for sums as a first step. This will leave you having to differentiate simpler expressions, and you can use the CTE on these, and so on...

Examples

1. (3x2-4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product.

2. (2x-1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient.

3. x2 + (4x-1)(x+2) can be computed by first calculating x2, then calculating the product (4x-1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum.

4. (3x2-1)5 can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power.

Using the Calculation Thought Experiment (CTE)

Let us use the CTE to find the derivative of

To use this method, pretend you were calculating, one step at a time, the value of for, say, x = 5. (You don't need to actually do the calculation.) One way of doing the calculation would be to use the following procedure:

Since the last operation is multiplication, the CTE tells us that the given expression is a product and so we should use the product rule.

Thus,

Remember that the expressions "d/dx" are short-hand for "the derivative of ..." In other words, we haven't done the work yet; the line above is just telling us what we need to do. (If we wanted, we could take a coffee break and come back to it later to do the work.)

To finish the calculation, we must compute the magenta- and blue-colored derivatives one-at-a-time and plug them in to the expression above:

The first (magenta) derivative is easy:

To calculate the second (blue) derivative, we need the quotient rule (use the CTE on the expression if you don't believe this...)

Now substitute these derivatives into formula (I) to obtain the answer:

Last Updated: July, 1999
Copyright © 1999 StefanWaner and Steven R. Costenoble