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Chapter 3 | Derivatives
dy dx ,
y ′, d
⎛ ⎝ f ( x )
⎞ ⎠ .
f ′( x ),
dx
dy dx | x = a
dy dx notation (called Leibniz notation) is quite common in
In place of f ′( a ) we may also use
Use of the
engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form Δ y Δ x where Δ y is the difference in the y values corresponding to the difference in the x values, which are expressed as Δ x ( Figure 3.11 ). Thus the derivative, which can be thought of as the instantaneous rate of change of y with respect to x , is expressed as dy dx = lim Δ x →0 Δ y Δ x .
Figure 3.11 The derivative is expressed as dy
Δ y Δ x .
dx = lim
Δ x →0
Graphing a Derivative We have already discussed how to graph a function, so given the equation of a function or the equation of a derivative function, we could graph it. Given both, we would expect to see a correspondence between the graphs of these two functions, since f ′( x ) gives the rate of change of a function f ( x ) (or slope of the tangent line to f ( x )). In Example 3.11 we found that for f ( x ) = x , f ′( x ) =1/2 x . If we graph these functions on the same axes, as in Figure 3.12 , we can use the graphs to understand the relationship between these two functions. First, we notice that f ( x ) is increasing over its entire domain, which means that the slopes of its tangent lines at all points are positive. Consequently, we expect f ′( x ) >0 for all values of x in its domain. Furthermore, as x increases, the slopes of the tangent lines to f ( x ) are decreasing and we expect to see a corresponding decrease in f ′( x ). We also observe that f (0) is undefined and that lim x →0 + f ′( x ) =+∞, corresponding to a vertical tangent to f ( x ) at 0.
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