A Partial Derivative Is the Rate of Change of a Multi-Variable Function When We Allow Only
Partial Differentiation
In the real world, it is very difficult to explain behavior as a function of only one variable.
Bound as we humans are to three spacial dimensions, multi-variable functions can be very difficult to get a good feel for. (Try picturing a function in the 17th dimension and see how far you get!)
We can at least make three-dimensional models of two-variable functions, but even then at a stretch to our intuition. What is needed is a way to cheat and look at multi-variable functions as if they were one-variable functions.
We can do this by using partial functions. A partial function is a one-variable function obtained from a function of several variables by assigning constant values to all but one of the independent variables. What we are doing is taking two-dimensional "slices" of the surface represented by the equation.
Example: can be modeled in three dimensional space, but personally I find it difficult to sketch!
But by alternately setting (red), (white), and (green), etc..we can take slices of (each one a plane parallel to the plane) and see different partial functions.
All of this helps us to get to our main topic, that is, partial differentiation. We know how to take the derivative of a single-variable function. What about the derivative of a multi-variable function? What does that even mean? Partial Derivatives are the beginning of an answer to that question.
When a function of more than one independent input variable changes in one or more of the input variables, it is important to calculate the change in the function itself. If we hold all but one of the variables constant and find the rate of change of the function with respect to the remaining variable, then this process is called partial differentiation
Partial derivative is the rate of change of a multi-variable function when we allow only one of the variables to change.
Specifically, we differentiate with respect to only one variable, regarding all others as constants
Which essentially means if you know how to take a derivative, you know how to take a partial derivative!
Geometrical Meaning
Suppose the graph of is the surface shown below. Consider the partial derivative of with respect to at a point .
Holding constant and varying , we trace out a curve that is the intersection of the surface with the vertical plane .
The partial derivative measures the change in per unit increase in along this curve. That is, is just the slope of the curve in direction at point. The geometrical interpretation of is 
analogous.
A partial derivative of a function f with respect to a variable x, say z=f(x,y1,y2,...yn) (where the yi's are other independent variables) is commonly denoted in the following ways:
(referred to as ``partial of z with respect to x'')
(referred to as ``partial of f with respect to x'' )
Note that this is not the usual derivative ``d'' . The funny ``d'' symbol in the notation is called ``roundback d'', ``curly d'' or ``del d'' (to distinguish from ``delta d''; the symbol is actually a ``lowercase Greek `delta', '').
The next set of notations for partial derivatives is much more compact and especially used when you are writing down something that uses lots of partial derivatives, especially if they are all different kinds:
Any of the above is equivalent to the limit
To get an intuitive grasp of partial derivatives, suppose you were an ant crawling over some rugged terrain (a two-variable function) where the x-axis is north-south with positive x to the north, the y-axis is east-west and the z-axis is up-down. You stop at a point P=(x0, y0, z0) on a hill and wonder what sort of slope you will encounter if you walk in a straight line north. Since our longitude won't be changing as we go north, the y in our function is constant. The slope to the north is the value of fx(x0, y0).
The actual calculations of partial derivatives for most functions is very easy! Treat every independent variable except the one we are interested in as if it were a constant and apply the familiar rules!
Example:
Let's find fx and fy of the function .
Second Partial Derivatives
Observe carefully that the expression fxy implies that the function f is differentiated first with respect to x and then with respect to y, which is a natural inference since fxy is really (fx)y.
For the same reasons, in the case of the expression,
it is implied that we differentiate first with respect to y and then with respect to x.
Below are examples of pure second partial derivatives:
Example:
Lets find fxy and fyx of
In this example fxy=fyx. Is this true in general? Most of the time and in most examples that you will probably ever see, yes. More precisely, if
· both fxy and fyx exist for all points near (x0,y0)
· and are continuous at (x0,y0),
then fxy=fyx.
Partial Derivatives of higher order are defined in the obvious way. And as long as suitable continuity exists, it is immaterial in what order a sequence of partial differentiation is carried out.
Try yourself!: Given , then find:
The symbol is usually read ‘the partial of with respect to , with held constant’. However, the important point to understand is that the notation means that has been written as a function of the variables and only, and then differentiated with respect to while being held constant.
Example: Let
Using polar coordinate and (), we can write in several ways. For each new expression let us find
In thermodynamics you will find these common expressions:
Added variables, same techniques
First, to define the functions themselves. We want to describe behavior where a variable is dependent on two or more variables. Every rule and notation described from now on is the same for two variables, three variables, four variables, and so on, so we'll use the simplest case; a function of two independent variables. Conventionally, z is the dependent variable (like y in univariate functions) and x and y are the independent variables (like x in univariate functions):
Basic rules of partial differentiation
The rules of partial differentiation follow exactly the same logic as univariate differentiation. The only difference is that we have to decide how to treat the other variable. Recall that in the previous section, slope was defined as a change in z for a given change in x or y, holding the other variable constant. There's our clue as to how to treat the other variable. If we hold it constant, that means that no matter what we call it or what variable name it has, we treat it as a constant. Suppose, for example, we have the following equation:
If we are taking the partial derivative of z with respect to x, then y is treated as a constant. Since it is multiplied by 2 and x and is constant, it is also defined as a coefficient of x. Therefore,
Therefore, once all other variables are held constant, then the partial derivative rules for dealing with coefficients, simple powers of variables, constants, and sums/differences of functions remain the same, and are used to determine the function of the slope for each independent variable. Let's use the function from the previous section to illustrate.
First, differentiate with respect to x, holding y constant:
Note that there were no y variables in the first term, so differentiation was exactly like the univariate process; in the last term there were no x variables, therefore the derivative is zero, according to the constant rule, since y is treated as a constant.
Now, take the partial derivative with respect to y, holding x constant:
Again, note that the first term had no "variables" in it, since x is being treated as a constant, therefore the derivative of that term is 0.
To make sure you have a clear picture of more than one slope in a function, let's evaluate the two partial derivatives at the point on the function where x = 1 and y = 2:
How do we interpret this information? First, note that when x = 1 and y = 2, then the function z takes on a value of 3. At this point on our "mountain' or 3 dimensional shape, we can evaluate the change in the function z in 2 different directions. First, the change in z with respect to x is 10. In other words, the slope in a direction parallel to the x-axis is 10. Now turn 90 degrees. The slope in a direction perpendicular to our previous slope is 6, therefore not quite as steep. Also, note that although each slope depends on the change in only one variable, the position or fixed value of the other variable does matter; since you need both x and y to actually calculate the numerical values of slope.But first, back to the rules.
The product and quotient of functions rules follow exactly the same logic: hold all variables constant except for the one that is changing in order to determine the slope of the function with respect to that variable. To illustrate the product rule, first let's redefine the rule, using partial differentiation notation:
Now use the product rule to determine the partial derivatives of the following function:
To illustrate the quotient rule, first redefine the rule using partial differentiation notation:
Use the new quotient rule to take the partial derivatives of the following function:
Try these!: If find
Not-so-basic rules of partial differentiation
First, the generalized power function rule. Again, we need to adjust the notation, and then the rule can be applied in exactly the same manner as before.
When a multivariate function takes the following form:
Then the rule for taking the derivative is:
Use the power rule on the following function to find the two partial derivatives:
The composite function chain rule notation can also be adjusted for the multivariate case:
Then the partial derivatives of z with respect to its two independent variables are defined as:
Let's do the same example as above, this time using the composite function notation where functions within the z function are renamed. Note that either rule could be used for this problem, so when is it necessary to go to the trouble of presenting the more formal composite function notation? As problems become more complicated, renaming parts of a composite function is a better way to keep track of all parts of the problem. It is slightly more time consuming, but mistakes within the problem are less likely.
The final step is the same, replace u with function g:
Special cases in multivariate functions
The last two special cases in multivariate differentiation also follow the same logic as their univariate counterparts.
The rule for differentiating multivariate natural logarithmic functions, with appropriate notation changes is as follows:
Then the partial derivatives of z with respect to its independent variables are defined as:
Let's do an example. Find the partial derivatives of the following function:
Try these!: If , find
The rule for taking partials of exponential functions can be written as:
Then the partial derivatives of z with respect to its independent variables are defined as:
One last time, we look for partial derivatives of the following function using the exponential rule:
Homework
1. For , find and at the points where
2. For verify a) b)
3. If ; find the following derivatives:
Power series in two variables
Taylor’s theorem for one independent variable
Taylor’s theorem expands in terms of , powers of and successive derivatives of , and can be stated as:
Where denotes the derivatives of
If and letting , we obtain Maclaurin’s series
Taylor’s theorem for two independent variables
If we consider where is a function of two independent variables and , then , in general, increases in and will produce a combined increase in .
For R : ………(1)
Where
From R to Q: is constant; changes to
………… (2)
Equation (1) and (2) can be manipulated to eventually lead to:
If the above equation can be written as:
Þ
Since and are small, the expression in the brackets and all subsequent expressions are very much smaller and can be discarded.
Dividing through by
The result above is known as total differential of
The results can be extended to three independent variables e.g
Example: A rectangular box has sides measured 30mm, 40mm and 60mm. If these measurements are liable to be in error by ± 0.5mm, ± 0.8mm and ± 1.0mm respectively, calculate the length of the diagonal of the box and the maximum possible error in the result.
If the sides of the box denoted by a, b and c
The diagonal
Kira sampai dapat!
Try to solve this problem:
The base radius r of a circular cone is increasing at the rate of 1.5 mm/s while the perpendicular height is decreasing at 6.0 mm/s. Determine the rate at which the volume V is changing when r = 12 mm and h = 24 mm