Partial Differentiation
Table of Contents
1. Definition
The principle of differentiation can be extended to functions of more than one variable.
The partial derivative of \( f(x, y) \) with respect to \( x \) is defined as:
\[ \frac{\partial f}{\partial x} = {f_x}\left( {x,y} \right) = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{(x + h) - x}\]
\( \fbox{\begin{minipage}{\dimexpr\textwidth-2\fboxsep-2\fboxrule\relax} The partial derivative of f(x, y) with respect to x is the rate of change of f(x, y) with respect to x , allowing \textbf{x} to vary but keeping \textbf{y} constant. \end{minipage}} \)
Similarly, the partial derivative of \( f(x, y) \) with respect to \( y \) is defined as:
\[ \frac{\partial f}{\partial y} = {f_y}\left( {x,y} \right) = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{(y + h) - x}\]
2. Interpretations
For a function of a single variable \( y = f(x) \), \( f'(a) \) can be interpreted as:
- The rate of change of \( y \) with repect to \( x \) at \( x = a \)
- The gradient of the tangent to the curve at \( x = a \)
Both of these can be applied to functions of multiples variables.
2.1. Classic Interpration - Rate of Change
\[ let\ z = f(x, y) \] Then \( {f_x}\left( {a,b} \right) \) represents the rate of change of \( z \) at \( x = a, \ y = b \) in the direction \( y = b \) (ie keeping \( y \) constant)
This interpretation follows nicely if we simply consider \( f(x, y) \) as a family of functions of \( x \), ie \( f(x) = x^2 + 2yx + y^2 \).
2.2. Geometric Interpretation - Tangent to surface
At any one given point \( P = (a, b, c) \) on the surface defined by \( z = f(x, y) \) there are an infinite number of lines tangent to the surface.
Using \( {f_x}\left( {a,b} \right) \) and \( {f_y}\left( {a,b} \right) \) we can find the equations of 2 lines tangent to the surface at \( P \). And hence the unique tangent plane to the surface at \( P \) (a plane is equivalent to the linear combination of two lines).
2.2.1. A Formula for the Tangent Plane
The formula for the tangent plane at \( (a, b, f(a, b) \) is given by: \[ t(x, y) = f(a, b) + f_x(x, y)(x - a) + f_y(x, y)(y - b) \] <b>Proof:</b> the tangent plane must satisfy the conditions:
\[ 1) \ t(a, b) = f(a, b) \] Direct substitution of \( (a, b) \) into the formula show this holds \[ 2) \ t_x(a, b) = f_x(a, b) \] \[ 3) \ t_y(a, b) = f_y(a, b) \] These must hold as the slope of the surface at \( (a, b) \) letting \( x/y = a/b \) be constant (ie \( f_x \) and \( f_y \) ), must be equivalent to gradient of the line produced by imposing this same constraint on the tangent plane.
3. Generalisation
Consider a surface in 3 dimensions. At any \( (x, y, z) \) point \( P \) on the surface there are an infinite number of lines tangent to that point (but only one tangent plane). However if we consider a plane passing through \( P \) (though not tangent), there is only one tangent line which lies on the plane and is tangent to the surface (considering the outline of the surface on the plane).
Consider the surface defined by \( f(x, y) = x^2y \)
Suppose we only consider the surface at some fixed \( y \), ie let \( y = b \) for some constant \( b \). This gives us a function of \( x, g(x) = bx^2 \).
Then \( g'(x) = 2bx \). The geometric interpretation of this derivative is that the tangent at \( (x, b, g(x)) \) on the plane \( y = b \), has gradient \( 2bx \). Hence equation of the tangent line given by \( y = b, x = z/2 \).