The Derivative Function If we replace a in the in the definition of the derivative the function f at the point x = a with a variable x, we get the derivative function f (x). Using Formula 2 gives f (x) 3 Once we have a formula for the derivative function, we can use it to find the value of the derivative at any point in the domain of the derivative function. The domain of the derivative function f must be a subset of the domain of the function f, since we cannot apply the definition of the derivative at an x-value where f is undefined. However, it is not guaranteed that the limit defining the derivative can be evaluated for all x in the domain of f. Clint Lee Math 112 Lecture 8: The Derivative 2/27 Differentiation We differentiate f (x) to find a formula for the derivative function f (x). In Example 27(c) we differentiated f (x) = x x 1 to find f (x) Clint Lee Math 112 Lecture 8: The Derivative 3/27
Example 28 Derivative of x 3 Let f (x) = x 3. (a) (b) Find f (x). Sketch the graphs of f (x) and f (x), and explain the relation between the two graphs. Solution (a): Using Formula 3 together with the binomial expansion gives f (x) = Note that f (1) as found in Example 22. Clint Lee Math 112 Lecture 8: The Derivative 4/27 Solution: Example 28(b) The graph of the function f (x) = x 3 looks like this. Now draw a short tangent line at a few points on the graph, say at x = 2, x = 1, x = 0, x = 1, and x = 2. The corresponding slopes are Now the graph looks like this. Now on a graph directly below the graph of f plot points for which the y-coordinate equals the slope at the corresponding point on the graph of f. Like this. Now join the points on the graph of the derivative. Like this. Clint Lee Math 112 Lecture 8: The Derivative 5/27
Solution: Example 28(b) Continued When visualizing the graph of the derivative of a function, remember that the y-coordinates on the graph of the derivative are determined by the slope at the corresponding point on the graph of the function. An important consequence of this is that the graph of the derivative does not depend on the vertical location of the graph of the function. So that, for example, the graph of the derivative of g(x) = x 3 + 5 will be identical to the graph of the derivative of f (x) = x 3. Clint Lee Math 112 Lecture 8: The Derivative 6/27 Example 29 Estimating Derivative of ln x Let f (x) = ln x. (a) Estimate the value of f (1). (b) Using your estimate from part (a) estimate the values of f (2), f (4), and f (10). (c) (d) Based on your estimates in parts (a) and (b) guess a formula for f (x). Sketch the graphs of f (x) and f (x) and explain how the graphs agree with your guess in part (c). Clint Lee Math 112 Lecture 8: The Derivative 7/27
Solution: Example 29(a) Part (a): Applying Formula 2 for the derivative at a point we have f (1) Since Further note that direct substitution gives 0/0 and that the factor and cancel approach is not an option. Thus, you should make a table of values to estimate the value of the limit. You should find lim h 0 ln(1 + h) h Clint Lee Math 112 Lecture 8: The Derivative 8/27 Solution: Example 29(b) Applying Formula 2 again we have f (2) Now applying the Laws of Logarithms gives ln(2 + h) ln 2 Then make the change of variables f (2) = 1 2 lim u 0 ln(1 + u) u in the limit to give In the same way, f (4) = and f (10) =. You do at least one of these yourself. Clint Lee Math 112 Lecture 8: The Derivative 9/27
Solution: Example 29(c) Based on the estimates in parts (a) and (b) we guess that f (x) Clint Lee Math 112 Lecture 8: The Derivative 10/27 Solution: Example 29(d) The graph of f (x) = ln x looks like this. The graph has positive slope for all x in the domain of f. With large positive slope for x near zero and decreasing slope as x increases. So the graph of f (x) looks like this. f (x) = ln x Clint Lee Math 112 Lecture 8: The Derivative 11/27
Solution: Example 29(d) Continued The graph of f (x) = 1 x f (x) = ln x shows positive, but decreasing, values for all x > 0, which is the domain of f (x) = ln x. However, note that the function g(x) = 1 x has domain (, 0) (0, ), which is not a subset of the domain of f (x) = ln x. This seems to contradict the requirement that the domain of the derivative is a subset of the domain of the function. We will return to this issue later. Clint Lee Math 112 Lecture 8: The Derivative 12/27 Example 30 Derivative of x Let g(x) = x. (a) (b) Find f (x). Sketch the graphs of g(x) and g (x), and explain the relation between the two graphs. Solution (a): Using Formula 3 gives g (x) = Multiplying top and bottom by gives g (x) = Clint Lee Math 112 Lecture 8: The Derivative 13/27
Solution: Example 30(b) The graph of the function g(x) = x looks like this. For x > 0, near x = 0, the slope is large and positive. As x increases from x = 0, the slope gets smaller, but stays positive. The graph never levels out, but the slope gradually approaches zero as x. The graph of g (x) looks like this. Clint Lee Math 112 Lecture 8: The Derivative 14/27 Example 31 Graph of the Derivative Function The graphs of four functions are shown. For each sketch the graph of the derivative of the function. (a) (b) g f (c) h (d) F Clint Lee Math 112 Lecture 8: The Derivative 15/27
Solution: Example 31(a) The graph of the function f, shown below, has positive slope for all x. For large negative x the slope is small and it becomes increasingly large as x increases. The graph of the derivative looks like this. f Clint Lee Math 112 Lecture 8: The Derivative 16/27 Solution: Example 31(b) The graph of the function g, shown below, has small negative slope for x large and negative. As x gets less negative, the slope initially becomes more negative, until it reaches its most negative value for an intermediate negative x. This is a minimum in the slope. After this point the slope becomes less negative again, until it goes to exactly zero at x = 0. As x increases from x = 0, the slope initially increases, getting more positive, until a maximum slope is reached for an intermediate positive x. After this the slope decreases but stays positive for all x > 0. It appears that the slope approaches zero as x ±. The graph of the derivative looks like this. g Clint Lee Math 112 Lecture 8: The Derivative 17/27
Solution: Example 31(c) The graph of the function h, shown below, has a slope that oscillates between positive and negative values. By the symmetry of the graph, it appears the maximum (most positive) and minimum (most negative) slopes will have the same absolute value. In particular, at x = 0, the graph has maximum slope. After x = 0 the slope decreases to zero and keeps decreasing until the graph crosses the x axis again, where it has its minimum value. The point where the slope is zero is exactly halfway between the two x-axis crossings, where the slope has its maximum and minimum values. After the slope reaches its minimum, it increases again, going through zero, and then reaching its maximum at the next x-axis crossing. After this, the graph repeats the same pattern. The graph of the derivative looks like this. h Clint Lee Math 112 Lecture 8: The Derivative 18/27 Solution: Example 31(c) Continued h You should recognize the graph of h as the graph of the graph of the derivative appears to be the graph of the verify this conjecture in a later lecture. function. The function. We will Clint Lee Math 112 Lecture 8: The Derivative 19/27
Solution: Example 31(d) The graph of the function F, shown below, has constant negative slope for x < 0 and constant positive slope for x > 0. Hence, the graph of the derivative looks like this. F If the x and y scales are the same, then the graph above is the graph of Clint Lee Math 112 Lecture 8: The Derivative 20/27 Solution: Example 31(d) F An important question is: What is F (0)? Using Formula 2 for the derivative at a point, we see that F (0) From our discussion in Example 3 we know that this limit Clint Lee Math 112 Lecture 8: The Derivative 21/27
Differentiability As seen in Example 31(d) a function that is continuous may have one (or more) points in its domain where the derivative does not exist. Definition: Differentiability at a Point We say that the function f is differentiable at a if exists and is finite. lim h 0 f (a + h) f (a) The graphs in Example 31(d) indicate that F(x) = x is h Clint Lee Math 112 Lecture 8: The Derivative 22/27 Example 32 Derivative of a Root Function Let F(x) = 3 x. (a) Find F (x). (b) Verify that F is not differentiable at x = 0 by direct use of Formula 2 for the derivative at a point. (c) Sketch the graphs of F(x) and F (x), and explain how they relate to the non-differentiability of F at x = 0. Clint Lee Math 112 Lecture 8: The Derivative 23/27
Solution: Example 32(a) Using Formula 3 we have F (x) Now change variable in the limit to Then, and So that F (x) Clint Lee Math 112 Lecture 8: The Derivative 24/27 Solution: Example 32(a) Continued Now using the factoring of the difference of cubes discussed in Example 22 we have F (x) Clint Lee Math 112 Lecture 8: The Derivative 25/27
Solution: Example 32(b) & (c) Using Formula 2 for the derivative at a point gives F (0) Since the limit is infinite, the function F is not differentiable at x = 0. The graphs of F(x) and F (x) are: The graph of F(x) has a the in the graph of F (x) at, which corresponds to Clint Lee Math 112 Lecture 8: The Derivative 26/27 Conditions for Nondifferentiability A function f is nondifferentiable at x = a if f is discontinuous at x = a the graph of f has an abrupt change in slope at a. This is called a cusp. the graph of f has a vertical tangent at a. Note that a function f cannot be differentiable at a point where it is discontinuous. However, if f is differentiable at a, then f must be continuous at a. A cusp is usually easy to see. However, they can be subtle, say if the slope changes from 1 on the left to 2 on the right of a point. A vertical tangent will always show up in the formula for the derivative, and, in most cases, it is apparent on the graph. Clint Lee Math 112 Lecture 8: The Derivative 27/27