# How to explain aspects of derivatives to little brother?

So as the title suggests, my $12$ year old little brother loves math. Since he is a bright kid, I started teaching him derivatives. The issue is he always keeps asking weird questions that are above his level. I'm trying to explain why you would use $\frac {dy}{dx}$ instead of $\frac {dx}{dy}$ when deriving $y=x^2$. I am particularly stuck on how to explain when to use those two. I keep telling him that you are finding the derivative of the function in respect to $x$, but he keeps asking why it is structured that way. Then I made the delta y over delta $x$ analogy, but he is still confused. I would greatly appreciate it if someone can find a visual way that I can explain when/how to use the latter of the two?

Edit: Sorry forgot to say, but he can't understand functions too, not even the slightest bit about function notation. So one can't use Lagrange's notation

• Why not just use a different notation, instead of getting hung up on this minor issue? – vadim123 Nov 25 '15 at 4:28
• Like $f'(x)$ for the function $f$. – Thomas Andrews Nov 25 '15 at 4:28
• $x$ is the independent variable. The derivative of $y$ with respect to $x$ answers the question, how does $y$ change if $x$ does? – John Douma Nov 25 '15 at 4:29
• Issue is he isn't getting function notation, so I have to stick with this one. – Michael Nov 25 '15 at 4:29
• if he doesn't want to know about function notation you've already found the issue. – Zelos Malum Nov 25 '15 at 4:30

The following is just my opinion. I commend you on trying to further your brothers education but I would be careful as to how far you go and in what order. In my experience, we tend to forget the things we knew before we learnt harder stuff. Given that your brother is asking about something notational, it may be that he doesn't really understand why you are doing any of the math to start with (especially if he hasn't been introduced to function notation yet).

Depending on what your brother already knows, I would suggest the following:

1) Teach him about linear functions and graphing them. This is usually the first introduction to the concept of slope (especially using gradient-intercept form). Hopefully your brother can already do this!

2) It would be useful if he knew how to graph quadratics (and cubics etc if possible). This will give you a broader range of functions with which you can give examples (and hence give pictures to aid your discussion of instantaneous rates of change).

3) Introduce some physical examples (e.g. motion under constant acceleration) to show why we are interested in rates of change at all.

If your brother can already handle these types of problems then perhaps you just need to use some of them more explicitly to show him what finding derivatives is all about. If he doesn't already know about the above three points then these may be better starting points for expanding his knowledge. Hopefully all this allows you to explain that the reason you are using $\frac{dy}{dx}$ is because you are interested in the rate of change of $y$ with respect to $x$ (rather than the rate of change of $x$ with respect to $y$). In a physical setting, for example, you are generally interested in the rate of change of distance with respect to time, rather than the rate of change of time with respect to distance. Hence we talk about velocity which is $\frac{dx}{dt}$ but there is no name for $\frac{dt}{dx}$.

• Thank you so much. I tried what you said and now he understand derivatives relatively well. Quite Well written answer. Would give you +1 if I could. – Michael Nov 25 '15 at 22:04