MATH 90 - The Derivative as a Function - Section 3.2 The derivative of f is the function f x lim h 0 f x h f x h for all x for which the limit exists. The notation f x is read "f prime of x". Note that the derivative "outputs" the slope of f at the point x, f x The process of finding the derivative is called differentiation A function is differentiable at x if its derivative exists at x. A function is differentiable on an open interval a, b if it is differentiable at every point in a, b Notations used for the derivative: f x, dy dx, D x y, y, d dx f x, D x f x The symbols dx d and D are called differential operators and should be thought of as functions whose domain and range are also functions. The notation dy is read as "the derivative of y with respect to x" (we say "dee-y dee-x"). This is the notation dx used by Leibniz when he presented the calculus in 684. Because of this it is often referred to as the Leibniz notation. For the function y f x, the idea of the Leibniz notation is dy dx lim x 0 y x lim x 0 f x x f x x f x When we want to state the value of a derivative using the Leibniz notation at a specific number a we will write dy dx x a
Examples:. Find the derivative of y t 2 Find the slope of the tangent line to y t 2 at the points 3, and 6, 2 Is this function differentiable at the point 2, 0? Interpret this graphically in terms of the tangent line at this point. 3 2 0 0 2 3 4 5 6 7 8 9 2
2. Find the derivative of g x 3 x Interpret g (solid line) and g (dashed line) graphically 8 7 6 5 4 3 2-8 -7-6 -5-4 -3-2 - 2 3 4 5 6 7 8 - -2-3 -4-5 -6-7 -8 3
3. A graphical example. Consider the graph of the function f shown below. Use it to sketch a graph of f on the same axes. 5 4 3 2-5 -4-3 -2-2 3 4 5 - -2-3 -4-5 4
Derivatives from the left and from the right The one-sided limit lim x a f x f a x a is called the derivative from the left of f at x a and the one-sided limit lim x a f x f a x a is called the derivative from the right of f at x a Note that f is differentiable at a if and only if the derivative from the right and from the left both exist and are equal. Differentiability & Continuity When might a function NOT be differentiable? If f is NOT continuous at x a then f is NOT differentiable at x a. This fact is usually stated in the following contrapositive form: If f is differentiable at x a, then f is continuous at x a Notice that this is NOT an if an only if statement. In other words, the converse of this theorem is NOT true. There are functions that are continuous, but NOT differentiable. The following two examples illustrate this. 5
. Consider the function f x x. The graph is shown below. 3 2-3 -2-2 3 Note that f is continuous everywhere (for all real numbers). So, in particular, it is continuous at x 0. However, note that lim x 0 f x f 0 x 0 x 0 lim x 0 x x lim x 0 x while lim x 0 f x f 0 x 0 x 0 lim x 0 x x lim x 0 x So that the derivative from the left and the derivative from the right are not equal. Hence, f is not differentiable at x 0. 6
2. Consider the function g x 3 x x /3. The graph is shown below. 2-8 -7-6 -5-4 -3-2 - 2 3 4 5 6 7 8 - -2 Note that g is continuous everywhere, including x 0. However, note that g x g 0 lim x 0 x 0 Thus, g is not differentiable at x 0. lim x /3 x 0 x lim x 0 x 2/3 These last two examples illustrate the main ways in which functions can be continuous at a point, but not differentiable at the point. The two cases can be described as. The function has a sharp corner. 2. The function has a vertical tangent line. Thus, in summary, a function is NOT differentiable at x a if. f is discontinuous at x a 2. f has a sharp corner at x a 3. f has a vertical tangent line at x a Summarizing the Relationship Between Differentiation & Continuity: Differentiability implies continuity BUT continuity does NOT imply differentiability. 7
For what values of x is the function h shown below not differentiable? 6 5 4 3 2-6 -5-4 -3-2 - 2 3 4-8
Graphical Interpretations of the Derivative Match the function on the left with its corresponding derivative on the right. 9
We now begin to develop rules for differentiating different functions using the limit definition. The Constant Rule: d dx c 0 for any constant c The Natural Exponential Function Rule Find the derivative of f x e x 0
The Power Rule Find the derivative of f x x n where n is any positive integer greater than. We will need the Binomial Theorem: x h x h x h 2 x 2 2xh h 2 x h 3 x 3 3x 2 h 3xh 2 h 3 x h 4 x 4 4x 3 h 6x 2 h 2 4xh 3 h 4 x h 5 x 5 5x 4 h 0x 3 h 2 0x 2 h 3 5xh 4 h 5 The coefficient pattern can be seen in "Pascal s Triangle": 2 3 3 4 6 4 5 0 0 5 There is a closed form formula for the binomial coefficients (see Algebra review sheet in front of book).
Find the derivative of the following functions:. y x 2 2. g t t 2.5 3. s x 4. h x x 4 5. d 3 u 5 2
Sum and Difference Rules & Constant Multiple Rule (Linearity Rules) If f and g are both differentiable functions and c any constant (i.e. real number) then f g, f g, and cf are all differentiable functions and d dx f x g x f x g x d dx f x g x f x g x d dx cf x cf x Proof: 3
Now we can differentiate any polynomial function: Find p x if p x 7x 3 5x 2 0x 4 4