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Partial differentiation - what is it about?

By Murray Bourne, 20 Jan 2009

Barbara, a reader from Mauritius, wrote recently:

What is the significance/logic behind partial derivatives? Why do we use it? I know it’s for functions involving 2 or more variables but the logic is not clear to me.

Multi-dimensional math is often hard to get your head around on the first hearing. The section on partial derivatives comes at the end of the introduction to the concept of derivatives. It would help if students have learned about 3-D geometry first.

Back to Barbara's mail. Here is my reply.

Hello Barbara

You could think of it like this. Say we are doing a car journey and we have no air conditioner in the car (so we notice the temperature).

The temperature in the car will depend on several things, but let’s restrict it to 3:

* How far north or south we are
* How high we are (altitude)
* The time of day

Let's now go from the tip of South America to the northern reaches of Canada, non-stop. We'll be going from the southern hemisphere across the equator to the northern hemisphere, up and down mountains, daytime and night-time.

So the temperature T will be a function of distance (x) from the equator, altitude (h) and time (t). In math-speak, we write this as:

T = f(x, h, t)

Now let’s consider partial dT/dx, which we write as ∂T/∂x. For this, we ignore height and time. The temperature will be low when we start (near the South Pole), get hot in the middle (as we pass the equator) and get cold again when we get to Canada. The graph of T against x will be bell-shaped and ∂T/∂x will be positive for the first half and negative for the second, something like the following.

bell-shaped curve

Next, consider partial ∂T/∂h. This time, we ignore x and t. Now the value of T will be higher on the coast than in the mountains, and will be up and down for the whole journey. Partial dT/Dh will vary between positive and negative throughout the journey.

Finally, ∂T/∂t. Keeping x and h constant, the temperature will be low in the mornings, higher in the middle of the day and then low again in the evenings. Partial dT/dt will be positive from dawn to around mid-day each day and negative for the rest of each day, something like our bell-shaped curve above.

So the idea of partials is to strip away all the other variables and just concentrate on one pair of variables at a time (the dependent variable and one of the independent variables). This is very important in any science experiment, as we need to test each variable to see what effect it is having on the overall picture.

Hope that helps.

Barbara replied:

Thanks for the mail. Really helpfull!!! Now i have a broader picture of how it works.

You’re welcome, Barbara.

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