# CA II.7 The Derivatives of cf and f+g

**Purpose**: To develop the general rule for taking derivatives of functions multiplied by a constant and functions added together. To introduce the differential operator and to develop an approach to checking derivative rules.

**Classroom Procedure**: This activity begins a series of activities (4 in total) developing the derivative rules that we use in class. The first page is where we introduce the differential operator. They should read this and try to make sense of it individually first and practice applying it at the top of the second page. Then talk to somebody nearby to compare their understanding and be ready to share this with the whole class. Once that is complete they then need to figure out how to develop an investigation in *Mathematica *to discover the derivative rule. This is a good thing to give time to here because if they understand how to build an investigation here it helps greatly in later activities. It is best if students can puzzle on this a bit for themselves and try to explain what their investigation is checking to each other. Then a whole class discussion on the form of the investigation and the results is beneficial after students have had time to work on this by themselves and become comfortable with it. Students then apply the rule in an application setting and are asked do they think it makes sense in that setting. After this we prove the rule from the definition of derivative (this could be done individually or as a class) and then apply it. This activity will probably take two days of class. If so then finishing Q4 in class and sending the students home with Q5 is a good breaking point. The second half of the activity should be worked through in the same way as above for the second rule.

**Ideas this Activity Builds On**: Students have already begun investigating whether a particular symbolic form of a derivative is justified by using a numerical and graphical approach (for e.g. when developing the power rule). This activity moves them to also seeing the ability to carry out a symbolic investigation using *Mathematica*. This could also be done with a graphing calculator or another CAS. This also builds on the power rule that they now know and can use.

**Introduction/Motivation of the Activity**: It is important to start each of these activities where derivative rules are being developed by highlighting why this is necessary to the students. One way to do this is to put a function on the board before each of these activities and ask if we know how to take the derivative of that type of function symbolically at this point. For this activity you could use f(x)=3x^2 or f(x)=x^2+x^3. It is also good to highlight the desire to figure out the derivative for any two functions added together or any function multiplied by a constant - not just these two.

**Need to Establish by the End of Activity/Wrap-Up**: The process of investigating a rule and how to take the derivative of a function times a constant multiple and two functions added together. Also the language of differential operator will be used again with no further introduction so it is important that students leave this activity comfortable with the meaning of this.

**Additional Notes**: In the synthesis questions here we purposefully have some optimization type questions to push students to apply their understanding of the derivative as well as their new ability to take derivatives symbolically of more complicated functions. We do this so that students do not get too far away from the meaning of the derivative while we are learning the derivative rules. We also want them to think about optimization throughout so that they have had time to become comfortable with it before hitting some of the more difficult problems in the third module. We also use a slightly different symbolic question at the end where students are given information about functions only at certain points and they need to use this information to find derivatives. This helps focus students on what each part of a "derivative rule" is really asking for instead of getting too caught up in a rote method without paying attention to what each piece is. We continue this through the product rule and chain rule as well. There should be in class time to work on CA II.7 Synthesis after students have tried them outside of class. There needs to be some discussion on their approach to the optimization questions in particular.