07-05-2012, 05:16 PM
DIFFERENCE BETWEEN PARTIAL DERIVATIVES AND TOTAL DERIVATIVES
PARTIAL DERIVATIVE.doc (Size: 266.5 KB / Downloads: 53)
Introduction
Suppose that ƒ is a function of more than one variable. For instance,
A graph of z = x2 + xy + y2. We want to find the partial derivative at (1, 1, 3) that leaves y constant; the corresponding tangent line is parallel to the x-axis.
It is difficult to describe the derivative of such a function, as there are an infinite number of tangent lines to every point on this surface. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the x-axis, and those that are parallel to the y-axis.
This is a slice of the graph at the right at y = 1.
A good way to find these parallel lines is to treat the other variable as a constant. For example, to find the tangent line of the above function at (1, 1, 3) that is parallel to the x-axis, we treat y as a constant one. The graph and this plane are shown on the right. On the left, we see the way the function looks on the plane y = 1. By finding the tangent line on this graph, we discover that the slope of the tangent line of ƒ at (1, 1, 3) that is parallel to the x axis is three. We write this in notation as
At the point (1, 1, 3),
or as "The partial derivative of z with respect to x at (1, 1, 3) is 3."
Definition
The function f can be reinterpreted as a family of functions of one variable indexed by the other variables:
In other words, every value of x defines a function, denoted FX, which is a function of one real number.[1] That is,
Once a value of x is chosen, say a, then f(x,y) determines a function fa which sends y to a2 + ay + y2:
In this expression, a is a constant, not a variable, so fa is a function of only one real variable, that being y. Consequently the definition of the derivative for a function of one variable applies:
The above procedure can be performed for any choice of a. assembling the derivatives together into a function gives a function which describes the variation of f in the y direction:
This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", "dah", or "partial" instead of "dee".
In general, the partial derivative of a function f(x1...xn) in the direction xi at the point (a1...an) is defined to be:
In the above difference quotient, all the variables except xi are held fixed. That choice of fixed values determines a function of one variable , and by definition,
In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.
An important example of a function of several variables is the case of a scalar-valued function f(x1...xn) on a domain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ∂f/∂xj with respect to each variable xj. At the point a, these partial derivatives define the vector
This vector is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f which takes the point a to the vector ∇f(a). Consequently the gradient determines a vector field.
Examples
The volume of a cone depends on height and radius
Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula
The partial derivative of V with respect to r is
It describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant.