![]() ![]() Times the derivative of the second function. In each term, we tookĭerivative of the first function times the second Plus the first function, not taking its derivative, ![]() Of the first one times the second function To the derivative of one of these functions, Of this function, that it's going to be equal Of two functions- so let's say it can be expressed asį of x times g of x- and we want to take the derivative If we have a function that can be expressed as a product Rule, which is one of the fundamental ways Personally, I don't think I would normally do that last stuff, but it is good to recognize that sometimes you will do all of your calculus correctly, but the choices on multiple-choice questions might have some extra algebraic manipulation done to what you found. If you are taking AP Calculus, you will sometimes see that answer factored a little more as follows: That gets multiplied by the first factor: 18(3x-5)^5(x^2+1)^3. ![]() Now, do that same type of process for the derivative of the second multiplied by the first factor.ĭ/dx = 6(3x-5)^5(3) = 18(3x-5)^5 (Remember that Chain Rule!) That gets multiplied by the second factor: 6x(x^2+1)^2(3x-5)^6 Your two factors are (x^2 + 1 )^3 and (3x - 5 )^6 Perfect practice makes perfect.Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. Generally, students can perform the product rule algorithm simplification of terms and algebraic manipulation is often the greatest challenge. Found on both the MC and FRQ sections of the test, students will be successful on these questions with consistent exposure to derivatives of products. This is a required skill that is tested on its own and as an intermediate step in more complex questions. The resulting derivative is intuitive and easy to remember! The concept of a limit is embedded in the notation! As a challenge for advanced learners, have students investigate the product of three functions, f(x)∙g(x)∙h(x). Explain to students that the Leibniz notation uses dr, dw, and dt to refer to a tiny, infinitesimal change. The regions in the diagram represents the change in photos, whereas question 4 gets at the rate of change these are related but not identical. Be aware of how the notation changes throughout the page. ![]() Be able to explain why the product of ∆r and ∆w is insignificant in this context. Review the formalization notes in the margin so you are able to clarify for students the meaning of all regions in questions 2 and 3. Students are provided a visual explanation of the product rule and are then asked to develop the product rule on their own. By itself, the product rule is generally not a challenge for calculus students, so we chose to make notational fluency, graphical representations, and connecting representations our focus today. The context for this lesson was interesting to our students and we had strong engagement in the lesson. Along with the derivative definitions and rules learned so far, the product rule is another foundational algorithm that students will use often throughout the AB and BC course. ![]()
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