The Calculation Thought Experiment (CTE)

Calculation Thought Experiment To decide which rule to use first in taking the derivative of an expression, we imagine that we are evaluating the expression for some value of x (by hand or with a scientific calculator), and note the last operation we would perform. This operation tells us which differentiation rule to use first. For instance, if it is multiplication, then we must use the product rule first, if it is addition, we must use the rule for sums first.

Using the Calculation Thought Experiment (CTE)

Let us use the CTE to analyze the expression

(3x+1)

x2+ 4 x2+x

.

To use the CTE, 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 follow the following procedure:

1. Calculate the quotient (x²+4)/(x² + x) = (25+4)/(25+5) = 29/30.
2. Calculate (3x+1) = 15+1 = 16
3. Multiply the answers from steps 1 and 2.

Since the last operation is multiplication, the CTE tells us that the given expression is a product .

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