Calculation Thought Experiment (CTE)
The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, difference, power...:
Given 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; if the last operation is raising to a power, treat the expression as a power, 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. (3x^{2}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. (2x1)/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. x^{2} + (4x1)(x+2) can be computed by first calculating x^{2}, then calculating the product (4x1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum.
4. (3x^{2}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.
