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In place of f ( a ) we may also use d y d x | x = a Use of the d y d x notation (called Leibniz notation) is quite common in 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 ( [link] ). Thus the derivative, which can be thought of as the instantaneous rate of change of y with respect to x , is expressed as

d y d x = lim Δ x 0 Δ y Δ x .
The function y = f(x) is graphed and it shows up as a curve in the first quadrant. The x-axis is marked with 0, a, and a + Δx. The y-axis is marked with 0, f(a), and f(a) + Δy. There is a straight line crossing y = f(x) at (a, f(a)) and (a + Δx, f(a) + Δy). From the point (a, f(a)), a horizontal line is drawn; from the point (a + Δx, f(a) + Δy), a vertical line is drawn. The distance from (a, f(a)) to (a + Δx, f(a)) is denoted Δx; the distance from (a + Δx, f(a) + Δy) to (a + Δx, f(a)) is denoted Δy.
The derivative is expressed as d y d x = lim Δ x 0 Δ y Δ x .

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 [link] we found that for f ( x ) = x , f ( x ) = 1 / 2 x . If we graph these functions on the same axes, as in [link] , 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 .

The function f(x) = the square root of x is graphed as is its derivative f’(x) = 1/(2 times the square root of x).
The derivative f ( x ) is positive everywhere because the function f ( x ) is increasing.

In [link] we found that for f ( x ) = x 2 2 x , f ( x ) = 2 x 2 . The graphs of these functions are shown in [link] . Observe that f ( x ) is decreasing for x < 1 . For these same values of x , f ( x ) < 0 . For values of x > 1 , f ( x ) is increasing and f ( x ) > 0 . Also, f ( x ) has a horizontal tangent at x = 1 and f ( 1 ) = 0 .

The function f(x) = x squared – 2x is graphed as is its derivative f’(x) = 2x − 2.
The derivative f ( x ) < 0 where the function f ( x ) is decreasing and f ( x ) > 0 where f ( x ) is increasing. The derivative is zero where the function has a horizontal tangent.

Sketching a derivative using a function

Use the following graph of f ( x ) to sketch a graph of f ( x ) .

The function f(x) is roughly sinusoidal, starting at (−4, 3), decreasing to a local minimum at (−2, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2).

The solution is shown in the following graph. Observe that f ( x ) is increasing and f ( x ) > 0 on ( 2 , 3 ) . Also, f ( x ) is decreasing and f ( x ) < 0 on ( , −2 ) and on ( 3 , + ) . Also note that f ( x ) has horizontal tangents at 2 and 3 , and f ( −2 ) = 0 and f ( 3 ) = 0 .

Two functions are graphed here: f(x) and f’(x). The function f(x) is the same as the above graph, that is, roughly sinusoidal, starting at (−4, 3), decreasing to a local minimum at (−2, 2), then increasing to a local maximum at (3, 6), and getting cut off at (7, 2). The function f’(x) is an downward-facing parabola with vertex near (0.5, 1.75), y-intercept (0, 1.5), and x-intercepts (−1.9, 0) and (3, 0).
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Sketch the graph of f ( x ) = x 2 4 . On what interval is the graph of f ( x ) above the x -axis?

( 0 , + )

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Derivatives and continuity

Now that we can graph a derivative, let’s examine the behavior of the graphs. First, we consider the relationship between differentiability and continuity. We will see that if a function is differentiable at a point, it must be continuous there; however, a function that is continuous at a point need not be differentiable at that point. In fact, a function may be continuous at a point and fail to be differentiable at the point for one of several reasons.

Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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