It is extremely easy to set up questions about differentiation techniques (but good real life application questions are really rare, esp. at secondary school level), apart from tedious computation, when the differentiation involves parameter, students may have difficulties, like mistaking:
Here is a question in recent quiz, which involves parametric equations:
Show that .
Quite a neat result, isn’t it?
Of course, if students could eliminate and obtain at the beginning, the solution path is easy.
But, many tried to find and and they gave
then, when they tried to differentiate the above directly, many were at a loss with variables and .
Not many students could move one step further to see the following “bridging" step:
Differentiate once more, yield .
How to set a question so as to obtain the neat result ?
Well, we may use the “walking backward" strategy, start with
We then solve the above differential equation
It is just an equation of circle, and it is not that difficult to re-write the equation into parametric form, like
A bit modification, we may consider an ellipse, say
and it is easy to have
Now, “walking backward", we can establish the parametric equation (one could choose as ugly as he likes) of the ellipse as
By the “set-in-advance" result
Differentiate with respect to , yield
Well, we may ask students to prove the above.
Urm, could we set up similar questions by considering the “set-in-advance" results, like
Students, try to explore it at your command.