Lecture 7(b) derivative as a function
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This document discusses the derivative as a function. It defines the derivative of a function f at a variable x as the limit of the difference quotient as h approaches 0. This allows the derivative to be viewed as a new function f'(x). The document also provides an example of sketching the graph of the derivative function f'(x) given the graph of the original function f(x). It demonstrates estimating the slope of the tangent line at various points to determine values of the derivative function. Finally, it discusses other notations used for the derivative, including D, d/dx, and dy/dx.
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Lecture 7(b) derivative as a function
- 1. The Derivative 2.7 as a Function
- 2. 2 The Derivative as a Function We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain
- 3. Example 1 – Derivative of a Function given by a Graph The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f ′. 3 Figure 1
- 4. Example 1 – Solution We can estimate the value of the derivative at any value of x by drawing the tangent at the point (x, f (x)) and estimating its slope. For instance, for x = 5 we draw the tangent at P in Figure 2(a) and estimate its slope to be about , so f ′(5) ≈ 1.5. 4 Figure 2(a)
- 5. 5 Example 1 – Solution This allows us to plot the point P ′(5, 1.5) on the graph of f ′ directly beneath P. Repeating this procedure at several points, we get the graph shown in Figure 2(b). Figure 2(b) cont’d
- 6. 6 Other Notations The symbols D and d/dx are called differentiation operators because they indicate the operation of differentiation.
- 7. 7 Other Notations The symbol dy/dx, which was introduced by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for f ′(x). We can rewrite the definition of derivative in Leibniz notation in the form
- 8. 8 Other Notations If we want to indicate the value of a derivative dy/dx in Leibniz notation at a specific number a, we use the notation which is a synonym for f ′(a).