9/1/2023 0 Comments Product rule calculus 2And then, let's see, three so plus three, X to theįourth times X squared is X to the sixth power. Sin squared of X times sin of X is sin of X to the third power. X to the fourth times X is X to the fifth. In which case, what would we get? Well let's see. And, if we want, we canĪlgebraically simplify. And then all of that isīeing multiplied by that. And then sin of X squared is sin squared of X. Them to the second power and then take their products. Raised to the second power I could take each of So all of this I could rewrite as let's see, this would be three times if I have the product Which actually, let me just rewrite that. That's just the product rule as applied to this part right over here. Plus the first expression X squared times the derivative of the second one. This is gonna be two X times the second expression sin of X. Well here, I would apply the product rule. Now, the second part, what would that be? The second part here do this in another color. Where the something, in this case, is X squared sin of X. So that would be three times that something squared times the derivative with respect to X of that something. So, if I take the derivative it would be the derivative with And so I have, I'm taking theĭerivative with respect to X of something to the third power. So I'll just say CR for chain rule first. And I encourage you to pause the video and see if you could work Interesting is that there's multiple ways to tackle it. Perfect practice makes perfect.Going to do in this video is try to find theĭerivative with respect to X of X squared sin of X. 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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