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Nicholas Kruse (A2), Apoorva Sinha (A4), Rhythm Shah(A6), Saumil Agarwal (A5)
Partial Derivatives
Algebraically, the best way of thinking of the partial derivative of a function with respect to x as the derivative of the function is to consider y a constant. Thus taking a partial derivative of any function is no more difficult than regular derivatives in Calc I.
Geometrically, the derivative with respect to x at a point P represents the slope of the curve that passes through P whose projection onto the xy plane is a horizontal line (Projections are a way of simplifying the function that essentially "removes" a dimension).
A few examples of simple partial derivatives:
'
' Note that the y variable is treated like a constant when differentiating with respect to x and vice versa with y
A visual representation of a partial derivative
The wire frame represents a surface, the graph of a function z=f(x,y), and the blue dot represents a point (a,b,f(a,b)). The colored curves are "cross sections" -- the points on the surface where x=a (blue) and y=b (green).
Recall the meaning of the partial derivative at a given point (a,b), the value of the partial with respect to x, i.e.
fx(a,b)
is the slope of the line tangent to the blue cross section. (Change in z over change in x.) In other words, it tells you how fast z changes with respect to changes in x.
The value of fy(a,b) shows the rate of change of z with respect to y. That's the slope of the line tangent to the green curve.
Second Order Partial Derivatives
Much like the second order derivatives of a one variable functions these derivatives have mathematically the same meaning though their calculations differ for functions of more than one variable
There are four different second order partials in the case of f(x,y):
However there is an important relationship between fxy and fyx:
Partial derivatives of functions of many variables
Suppose that
f(x,y,z) = xy - 2yz
is a function of three variables, then we can define the partial derivatives in much the same way as we defined the partial derivatives for three variables.
We have
- fx = y
- fy = x - 2z
- fz = -2y
Second order partials still function in the same way except with more possible combinations however the theorem above shows that the order of derivation does not matter upon these.
Hopefully, this website was helpful in providing you with insight into partial derivatives, their meaning as well as the method used to solve for them.
