Experience First

After spending time developing the conceptual framework of a derivative, we now move towards more efficient strategies for calculating these derivatives. In today’s lesson, students reason about the derivative of constant function and a linear function and generalize their findings to all constant and linear functions. In question 3, students notice patterns in the derivatives of power functions, namely that the exponent becomes the coefficient of the derivative and the power is lowered by one. This becomes one of the most useful derivative shortcuts the students will use in Calculus!

 

In the rest of the activity, students reason about the constant multiple rule and the sum and difference rule from a graphical perspective. How does shifting a graph vertically affect the slope? (It doesn’t).  How does a vertical stretch of k affect the slopes? (All the slopes are now k times as steep). Putting these ideas together will help students find the derivative of any polynomial. 

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Formalize Later

Students generally catch on to the derivative shortcuts fairly easily. The struggle tends to emerge when we deal with rational and negative exponents. A quick review of Algebra concepts may be necessary (namely that a square root is a ½ power, or that dividing by x^4 is the same as multiplying by x^-4). As time allows you can explore these two properties in as great of depth as you choose. We tend to value students understanding these properties more so than just their ability to memorize the “rules”. To explain the square root property, we talk about how squaring the square root results in just x, and raising x^(½) to the 2nd power, also results in just x, so the two expressions must be equivalent. To explain the negative exponent property, we use a table to show the powers of 3, for example. Multiplying by 3 increases the power of 3 by 1, whereas dividing by 3, lowers the power by one. Thus, when dividing by 3^4, we are dividing by 3 four times, resulting in 4 less powers of 3, as denoted by 3^-4.

 

When finding the derivative of a polynomial, students generally struggle with the linear term the most. Many students say the derivative of 3x is either 3x or 0, instead of seeing this term as a linear function whose slope is always 3.