Differentiable functions

This idea goes back to Barrow who constructed tangent lines as limits of secants (see Barrow's differential triangle).

For functions of more than one variable, say f:R² -> R such a limit is unlikely to exist since the slope of a tangent line would naturally depend on how you approach a point. If you consider secants between (x, y) and (x+dx, y+dy) for small dx and dy, their slopes would normally vary quite a bit depending on the relative sizes of dx and dy, i.e. on direction of the vector (dx, dy).

What we really want then is a generalization of the tangent line to the tangent plane (or a higher dimensional linear manifold for functions of more variables).

What does tangent mean?

The calculus answer is through linear approximation. The tangent line should be a ``good'' linear approximation to f near x.

For functions of, say, 2 variables the tangent plane should be a ``good'' linear approximation to f near (x, y).

What does ``good'' mean?

Good means the graph approaches the tangent line like a parabola approaches the x axis near the origin - really fast. In other words, the error of approximation gets small ``really'' fast.

Linear approximation

If L not vertical, L(x+dx)=f(x)+m dx, where m is its slope.

The error between L and f is = L(x+dx)-f(x+dx) = f(x)+m dx-f(x+dx).

As dx -> 0, if f(x+dx) -> f(x), i.e. if f is continuous at x, then -> 0.

To see how fast goes to 0 compared to dx, we can take the ratio of the two /dx = m-[f(x+dx)-f(x)]/dx.

We can say that goes to zero ``faster'' than dx whenever the ratio between them goes to 0 as dx -> 0.

Definition: We say that f is differentiable at x, if /dx -> 0 as dx -> 0.

(For functions of one variable this is equivalent to the previous definition.)

If /dx -> 0 as dx -> 0, then m=f'(x), and with this slope L is exactly the tangent line L(x+dx)=f(x)+f'(x) dx.

Thus, f(x+dx)=f(x)+f'(x) dx+.

Definition: The function f'(x) dx is linear in dx and is called the differential of f (written df).

The same idea works for functions of several variables.

Suppose f: R² -> R and L is a plane through the point (x, y, f(x, y)).

If L is not vertical, L(x+dx, y+dy)=f(x, y)+m dx+n dy, where m and n are the slopes in the x and y directions.

The error is =L(x+dx, y+dy)-f(x+dx, y+dy)=f(x, y)+m dx+n dy-f(x+dx, y+dy).

To express the notion of tangency we look at the ratio of error to the magnitude of displacement dr=|(dx, dy)|.

Definition: We say that f is differentiable at (x, y), if /dr -> 0 as dr -> 0.

We can determine the values of m and n in the equation for L which make L tangent to the graph of f by alternately letting x or y stay consant, i.e. by letting the displacement (dx, dy) approach (0, 0) along horizontal or vertical directions.

(For functions with more variables, keep all but one variable constant at a time.)

If y is constant, i.e. dy=0, then dr= dx, so /dr=m-[f(x+dx, y)-f(x, y)]/dx.

If /dr -> 0, then m=lim [f(x+dx, y)-f(x, y)]/dx, which is known as the partial derivative of f with respect to x (written f_x).

Similarly, by keeping x constant we can deduce that n=f_y(x, y).

Thus, f(x+dx, y+dy)=f(x, y)+f_x(x, y) dx+f_y(x, y) dy+.

Again the differential df is the linear function of dx and dy:

df=f_x dx+f_y dy=(f_x, f_y)·(dx, dy).

Definition: The vector (f_x, f_y) is called the gradient of f (written f).

We may write the formula for df also in matrix notation:

df = [ fx fy ] [ dx ]
               [ dy ]

Matrix notation is particularly useful when the range of f is not just R but a higher dimensional space, say R². In this case the derivative matrix has more rows - gradients of the components of f.

Theorem: If the partial derivatives of f exist and are continuous in a neighborhood of (x, y), then f is differentiable at (x, y).