Estimating the Derivative at a Point on the Graph of a Function

To find the functional value, $f\left(a\right),$ find the y-coordinate at $x=a.$

To find the derivative at $x=a,$ $f^{\prime }\left(a\right),$ draw a tangent line at $x=a,$ and estimate the slope of that tangent line. See Figure 6.

Figure 6

1. ⓐ $f(0)$ is the y-coordinate at $x=0.$ The point has coordinates $\left(0,1\right),$ thus $f(0)=1.$
2. ⓑ $f(2)$ is the y-coordinate at $x=2.$ The point has coordinates $\left(2,1\right),$ thus $f(2)=1.$
3. ⓒ $f^{\prime }(0)$ is found by estimating the slope of the tangent line to the curve at $x=0.$ The tangent line to the curve at $x=0$ appears horizontal. Horizontal lines have a slope of 0, thus $f^{\prime }(0)=0.$
4. ⓓ $f^{\prime }(2)$ is found by estimating the slope of the tangent line to the curve at $x=2.$ Observe the path of the tangent line to the curve at $x=2.$ As the $x$ value moves one unit to the right, the $y$ value moves up four units to another point on the line. Thus, the slope is 4, so $f^{\prime }(2)=4.$