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 the temperature drops. Let's denote the average temperature by and define a function where is the heating bill as a function of .
We can then interpret the ordinary derivative as indicating how much the heating bill will change as you change the temperature:
If we plot as a function of then gives the slope of the graph at the point where . We say that is the derivative of with respect to . If is given in degrees Celsius then is the change in the heating cost per degree Celsius of temperature increase when the temperature is . Since decreases as increases, we would 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 ).
Example
In this example we have a curve:
where is the average temperature in degrees Celsius.
The derivative of this curve is:
The derivative gives us the slope of the curve at any point. Since we would like to plot the tangent of the curve where , we plug that in and get the slope of the tangent line .
To find the y-intercept of the tangent line, we first find the value of at :
We can find then the y-intercept by using the formula:
Therefore:
Partial derivatives are analogous to ordinary derivatives
Writing the heating bill as a function of temperature is a gross oversimplification. The heating bill will depend on other factors. For instance, the amount of insulation in your house, which we'll denote by . We can define a new function where gives the heating bill as a function of both temperature and insulation .
Suppose you aren't changing the amount of insulation in your house so we view 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 as a function of both and , and we are explicitly leaving one of the variables () constant. In this case, we call the change in the partial derivative of with respect to , a term that reflects the fact some variables remain constant. We also change our notation by writing the as a , so that
If is given in degrees Celsius, then is the change in heating cost per degree Celsius of temperature increase when the outside temperature is and the amount of insulation is .
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 with respect to :
If is given in centimetres of insulation, then is the change in heating cost per added centimetre of insulation when the outside temperature is and the amount of insulation is .
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 , the partial derivatives can be viewed as the slopes of the graphs in the direction and in the direction.
Examples of calculating partial derivatives
Once you understand the concept of a partial derivative as the rate at which something is changing, calculating partial derivatives usually isn't difficult. (Unfortunately, there are special cases where calculating the partial derivatives is hard.) As these examples show, calculating a partial derivative is usually just like calculating an ordinary derivative in one-variable calculus. You just have to remember which variable you are taking the derivative of.
Example 1
Let . Calculate
Solution:
We view as being a fixed number and calculate the ordinary derivative with respect to . becomes so we are left with:
Example 2
Also for . Calculate
Solution:
Because this time we are finding the derivative with respect to we treat as a constant. becomes so we have:
Example 3
Also for . Calculate
Solution:
Example 4
For
calculate
Solution:
Although this initially looks hard, it's an easy problem. The ugly term does not depend on , so in calculating the partial derivative with respect to , we treat it as a constant. The derivative of a constant is 0, so that term drops out. The derivative is just the derivative of the last term with respect to , which is
Substituting the values , we obtain the final answer
Example 5
Given
calculate at the point
Solution:
In calculating partial derivatives, we can use all the rules for ordinary derivatives. We can calculate using the quotient rule:
TODO: finish this example