Introduction to partial derivatives
Ordinary derivatives in one-variable calculus
Your heating bill depends on the average temperature outside. If all other factors remain constant, then the heating bill will increase when temperatures drop. Let’s denote average temperature by T, and define a function h : R → R so that h(T) gives the heating bill as a function of T.
We can then interpret the ordinary derivative (i.e., the derivative you learned about in first semester calculus) as indicating how much the heating bill will change as you change the temperature:
 (a) =  ( at T = a). |
(a) gives the slope of the graph at the
point where T = a. We say that 
is the derivative of h with respect to T. If T
is given in degrees Celsius, then 
(a) is change in heating cost per degree
Celsius of temperature increase when the temperature is a. Since h decreases as T
increases, we expect 
to be negative. (The rate of change in heating cost per
degree of Celsius of temperature decrease is positive. But this positive rate is
equal to -
.)
Although I don’t know what h(T) should really look like, pretend it looks
like thick green curve graphed in the below CVT. The point (a,h(a)) is
shown in red, which you can change by dragging with your mouse. We can
visualize the derivative by drawing a (thin blue) line tangent to the curve
at the point (a,h(a)). The slope of the line is equal to the slope of the
graph when T = a; hence, the slope of the line is equal to the derivative
(a).
Partial derivatives are analogous to ordinary derivatives
Clearly, writing the heating bill as a function of temperature is a gross oversimplification. The heating bill will depend on other factors, not least of which is the amount of insulation in your house, which we’ll denote by I. We can define a new function h : R2 → R so that h(T,I) gives the heating bill as function of both temperature T and insulation I.
Who knows, maybe this function would look something like the below graph.
Suppose you aren’t changing the amount of insulation in your house, so that we view I as a fixed number. Then, if we look at how the heating bill changes as temperature changes, we’re back to our first case above. The only difference is that we now view h as a function of both T and I, and we are explicitly leaving one of the variables (I) constant. In this case, we call the change in h the partial derivative of h with respect to T, a term that reflects the fact some variables remain constant. We also change our notation by writing the d as a ∂, so that
 (a,b) =  ( at T = a while holding I constant at b). |
(a,b) is change in heating cost per degree
Celsius of temperature increase when the outside temperature is a and the
amount of insulation is b.
Now, imagine you are considering the possibility of lowering your heating bill by installing additional insulation. To help you decide if it will be worth your money, you may want to know how much adding insulation will decrease the heating bill, assuming the temperature remains constant. In other words, you want to know the partial derivative of h with respect to I:
 (a,b) =  ( at I = b while holding T constant at a). |
(a,b) is change in heating cost
per added centimeter of insulation when the outside temperature is a and the
amount of insulation is b.
The partial derivative 
indicates how much effect additional insulation will
have on the heating bill. Since additional insulation will presumably lower
the heating bill, 
will be negative. If additional insulation will have a
large effect, then 
will be a large, negative number. If, for your house,
is large and negative, you may be inclined to add insulation to save
money.
In the graph of h(T,I), the partial derivatives can be viewed as the slopes of the
graphs in the T direction and in the I direction. In the below CVT, the partial
derivative 
corresponds to the slope of the dark blue line, and the partial
derivative 
corresponds to the slope of the light green line. (The numerical
values of these partial derivatives are also displayed. However, I wrote
them like dh∕dT because I couldn’t get a ∂ character to display in the
graph.)
You can drag the red point around to change the values of T and I see how, for
example, the partial derivative 
depends on both temperature and insulation.
Consequently, your decision to add insulation will be affected by what
temperatures you expect and how much insulation your home has already. You
might expect that additional insulation will have a larger effect (i.e., 
will be
larger negative number) for lower temperatures and smaller amounts of insulation.
So if you live in Minnesota and have an old, poorly insulated house, it’s
likely that 
will be a very large, negative number so that adding a
moderate amount of insulation could dramatically decrease your heating
bill.
As mentioned above, a heating bill depends on many more factors than
temperature and insulation. We could define a function h of many variables to
give a more accurate estimate of heating costs. However, the math is the same no
matter how many variables h depends on, as long as it depends on two or
more. So, if you want to include the effect of the size S of a house, you
can define h(T,I,S) to be the heating bill as a function of temperature,
insulation, and size. Then 
(T,I,S) would tell you how much the heating
costs change as you change the size, leaving temperature and insulation
constant. (Even though I cannot plot a graph of h(T,I,S), I hope that the
concept of the partial derivatives of h still makes sense.) I imagine the
value of 
would be useful only if you are planning to move since you
probably don’t plan to cut a room off your house to save heating costs
(though I suppose you could just not heat a room and effectively reduce the
size).
I hope you stay warm this winter.
You can find examples of calculating partial derivatives by following this link.
