Differentiability

Just as we have defined the derivative of a function at a point, we can define the derivative (or derived function) of a function f(x) as

df f(x + ∆x) − f(x) f 0(x) = = lim dx ∆x→0 ∆x The domain of f 0(x) is all points in the domain of f(x) for which the derivative is defined at that point. Similarly, we could say that the domain of f 0(x) is the domain of f(x) minus all of the points for which the derivative is not defined. The derivative is not defined in following situations.

1. Any point where the function f(x) is not defined. If f(x) is not defined at x0, surely it makes little sense to talk about the derivative at x0 2. Any point where the function f(x) is not continuous. At both point and jump discon- tinuities, f 0(x) is not defined.

3. Any point where the function f(x) is not smooth. If f(x) fails to be smooth at a point, it means there is either a corner or a cusp. At a corner the left and right-hand limits constituting the derivative are not equal; at a cusp one is ∞ and the other is −∞.

4. Any point where the function f(x) has a vertical tangent. In this case, the slope approaches ±∞, so the slope is undefined.

Linear functions are one of the most simple functions to find the derivative of. Based on the above criteria, linear functions are always differentiable at all points in their domain. If we consider two points on the line and draw a secant line through them, it is exactly the slope of the line. No matter how close or far apart the two points are, the slope is the same (the average rate of change of a linear function is constant). Thus, if we take the limit as the distance between the two points approaches 0 (in order to find a tangent line) it will simply have the same slope as the line itself. It follows that if we have a line with the equation

f(x) = mx + b we find df f 0(x) = = m dx We can verify the above argument by applying the definition of the derivative to the above function f(x).

f(x + ∆x) − f(x) m(x + ∆x) + b − (mx + b) m∆x f 0(x) = lim = lim = lim = lim m ∆x→0 ∆x ∆x→0 ∆x ∆x→0 ∆x ∆x→0 Thus f 0(x) = m just as in the above argument.

Example 1 Suppose A(f) = 0.5f − 10. Find A0(f). Solution Since A(f) is simply a linear function, we know that A0(f) = 0.5, the slope of the line.

Example 2 Suppose T (m) = 1 − 2m. Find T 0(m). Solution Since T (m) is simply a linear function, we know that T 0(m) = 2, the slope of the line.

Example 3 Suppose F (t) = 3 · 107. Find F 0(t). Solution Since F (t) is a constant function, it is a line with slope 0. Thus, F 0(t) = 0.

Another basic type of function we can calculate the derivative of is a quadratic function. Recall the falling basketball with position given by

y(t) = 9.8t2

We previously found the derivative at a point t0, which gave us a derived function that was a function of t0. Simply substituting t for t0 we find the derivative as a function of time is

y0(t) = 18.6t We can also rewrite the above equation as dy = 18.6t dt which is a differential equation. We already know that y(t) = 9.8t2 is a solution to the dif- ferential equation, because that was the equation we used to derive the differential equation. If we didn’t know the solution, we’d need to use integration, or one of the other methods used to solve differential equations.

The derivative of a function describes the instantaneous rate of change of the function. Consider the following 3 cases.

1. The derivative is positive. This implies that the instantaneous rate of change of the function is positive, which means the function is increasing.

2. The derivative is negative. This implies that the instantaneous rate of change of the function is negative, which means the function is decreasing.

3. The derivative is zero. This implies that the instantaneous rate of change of the function is zero, which means the value of the function is constant (not changing). The derivative tells us more than just in what way the function is changing, but also to what extent. If the magnitude (absolute value) of the derivative is large, then the function is either increasing or decreasing rapidly. If the magnitude of the derivative is small, then the function is either increasing or decreasing slowly. If the magnitude of the derivative is zero, then the function is not changing at all.

Using the above information, we should be able to create a reasonable sketch of a function from its derivative, or a reasonable sketch of the derivative from the function itself.

Example 4 Sketch the function that is associated with the following graph of its derivative.

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-20 -10 -5 0 5 10 x Solution We will use the information provided by the derivative in order to sketch the associated function. At first, the derivative is negative with a large magnitude, which means the function is decreasing rapidly. As the magnitude of the derivative decreases, so does the rate at which the function decreases. When the derivative crosses zero the function stops decreasing and begins increasing slowly. Afterwards the derviative continues to increase, so the function begins to increase more rapidly. A function with these characteristics is the familiar parabola, shown below.

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0 -10 -5 0 5 10 x It is important to note that although the derivative gave us information of how the function looks, it does not tell us if the function is shifted vertically or not; the parabola may be shifted upwards or downwards. Note that such a shift would be represented by adding a constant to the parabola. Now if we were to take the derivative of a function that looked like

x2 + c where c is a constant, the constant term c would vanish. Thus, the derivative will never tell us whether or not the function is shifted vertically, and likewise, functions that may be shifted vertically from each other have the same derivative. This is because the derivative only tells us the rate of change of a function, which does not change with the addition of a constant term.

Unless the derivative is constantly zero, points where the derivative is zero correspond to points where the derivative is either changing from positive to negative, or negative to positive. In terms of the function itself, these points correspond to places where the function either stops increasing and begins decreasing, or stops decreasing and begins increasing. This observation leads us to the following definition.

Definition: Critical Point A function f has a critical point at x in the domain of f if f 0(x) = 0 or if the derivative is not defined at x.

Note that the definition of critical point also includes points where the derivative is not defined. The significance of critical points, as we will see later, is that they are points where a local minimum or maximum of the function may occur (it doesn’t necessarily have to occur there).