Clearly, the student knows that derivatives have *something* to do with tangent lines, he/she is just not sure what. I would want to stress the meaning of derivative as the limit of the difference quotient (aka slope). I would also remind this student that in order to write the equation of a line, two things are needed: a point and a slope. Until you have both, you do not have all the necessary requirements.

mr bombastic

It is frustrating how often I get evidence, like this, that reveals a complete lack of understanding of a most basic fundamental idea.

Addressing the issue the student had with this problem is more or less irrelevant. Some how you need to get through to this student that it is important that they understand what they are doing. Can you make a quick sketch or use some mental math to see if your answer makes sense? Do you understand and can you give examples for the vocabulary we are using?

Often you need to have a student 1-1 to really get a good sense of their understanding and misunderstanding. I need to do a better job of having a larger portion of assessments really checking for understanding, which is difficutl to do. On the other side of the coin, students need to take much more responsibility for their understanding.

I’d say this is a fairly common type of error, confusing a derivative (a FORMULA for a slope of a tangent line) with the slope itself or with the tangent line. bombastic makes a solid point that this is about fundamental understandings. Ignoring the second = in line 1 where the student was attempting to rewrite f as a power function, I think it’s important to note that the student’s error mostly is one of omission, not commission. This was good work, just not enough of it.

I argue that the student could have known that s/he was in error, and perhaps that would be a good way to have students work their own way out of this. When I’ve encountered this, I ask the student if the provided answer is an equation of a line and leave him/her to reason it out. Some take longer than others to understand what their answer actually says and how that is different from what was asked. Given the length of many calculus problems, I encourage my students to re-read the question after they have finished to be sure their answer matches what the question actually asked.

If you really want to test for the presence of this misunderstanding, give a 3-term quadratic function and ask for an equation of a tangent to a point. That the first derivative is itself a linear function catches many students who aren’t paying full attention.