Specify a function \(f(x)\). Then use the sliders to choose a value for \(c\) and \(h\)
to explore how the derivative is defined through a limit.
Function: \(f(x) = \)
The derivative of a function \(f(x)\) at a point \(c\) is defined by a limit \[ \left. \frac{df}{dx} \right|_{x=c} = \lim_{h \to 0} \frac{f(c+h)-f(c)}{h}. \] This definition is illustrated using the graphs above.
Once you define your function, choose a point \(x=c\) by moving the red point on the x-axis of the graph on the left. There is a corresponding point on the graph \(y=f(x)\) at this point, and the y-value represents the value of \(f(c)\).
There is also a second point on the x-axis that is defined as \(x=c+h\). To move this point, you need to adjust the value of \(h\) by moving the blue point on the x-axis of the graph on the right. The value of \(h\) determines the location of the second point relative to the value of \(c\). The y-value of the graph on the left at this point represents \(f(c+h)\).
A line (orange) is drawn between the two points \((c,f(c))\) and \((c+h,f(c+h))\), which is called the secant line. (The line segment joining the points is sometimes called the chord.) This line has a slope, and the slope depends on both \(c\) and \(h\). The graph on the right plots the value of the slope as it depends on \(h\) for the given value of \(c\).
When \(h=0\), there are not two points to calculate the slope. So the secant line disappears in the figure on the left, and the curve on the right has a hole when \(h=0\). However, it is clear for most functions that the formula for the slope coming from the left or the right has a limiting value.
The derivative is this limit. In other words, the y-value of that hole at \(h=0\) in the graph on the right is the derivative of the function graphed on the left at the particular point \(c\) that is being explored. Each value of \(c\) has a different limiting slope, corresponding to the derivative at that point. This means that the derivative is a function of the value \(c\).