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, or power:
Given an expression, consider the operations you would use in computing its value following the standard order of operations. 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 two simpler expressions, and you can then use the CTE on these, and so on...
#[Examples][Ejemplos]#
#[
1. $(3x^2- 4)(2x+1)$ is calculated by first calculating the expressions in parentheses and then multiplying. Since the last step is
multiplication, we treat the expression as a
product.][
1. $(3x^2- 4)(2x+1)$ se calcula primero calculando las expresiones entre paréntesis y luego multiplicándolas. Como el último paso es
multiplicación, tratamos la expresión como un
producto.]#
#[
2. $\dfrac{2x-1}{x}$ is calculated by first calculating the numerator and denominator separately, and then dividing one by the other. Since the last step is
division, we treat the expression as a
quotient.][
2. $\dfrac{2x-1}{x}$ se calcula calculando primero el numerador y el denominador por separado, y luego dividiendo uno por el otro. Como el último paso es
división, tratamos la expresión como un
cociente.]#
#[
3. $(4x-1)(x+2) + x^2$ is calculated by first calculating the product $(4x-1)(x+2)$, then calculating $x^2$, and finally adding the two answers. Since the last step is
addition, we treat the expression as a
sum.][
3. $(4x-1)(x+2) + x^2$ se calcula calculando el producto $(4x-1)(x+2)$, luego calculando $x^2$, y finalmente sumando las dos respuestas. Como el último paso es
sumar, tratamos la expresión como una
suma.]#'
#[
4. $(3x^2-1)^5$ is calculated by first calculating the expression in parentheses, and then raising the answer to the fifth power. Since the last step is
raising to a power, we treat the expression as a
power..][
4. $(3x^2-1)^5$ se calcula calculando la expresión entre parénteses, y luego elevar la respuesta al quinto potencia. Como el último paso es
elevar a una potencia, tratamos la expresión como una
potencia.]#