Worksheet for Section 2.3

Section 2.3 takes the next step toward a general approach to slopes of tangent lines, treating the derivative as a function -- define the derivative of $f(x)$:

$$f'(x)=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

Note that the derivative of $f(x)$ is itself also a function of $x$ -- the idea here is that the slope of the tangent line at $x=c$ is equal to $f'(c)$, so that to find slopes of tangent lines at different points on the graph, you simply have to plug different $c$ values into the function $f'(x)$. This limit may not exist -- for example, if the function has a vertical tangent line, or if the one-sided limits take different values. To see how this definition of a derivative works, use it to compute derivatives of the following functions:

$$f_1(x)=x^2\qquad f_2(x)=x^3-3x\qquad f_3(x)=\sqrt{x}\qquad f_4(x)=\frac{2}{x}$$

With the function $f_1(x)$, use your derivative function to compute $f'(0)$, $f'(1)$, and $f'(2)$. Verify these both graphically (by using the values as slopes of tangent lines and graphing the lines) and numerically (by generating tables of values). Using the derivative you computed for $f_3(x)=\sqrt{x}$, find an equation for the tangent line to the graph of $f_3(x)$ at each of the points $(1,1)$ and $(4,2)$ on the graph.

Compute the derivatives of $g_1(x)=x^3$ and $g_2(x)=x^4$. These, along with the derivative of $f_1(x)=x^2$ above, suggest a pattern for derivatives of powers of $x$: if $f(x)=x^n$, then $f'(x)=n x^{n-1}$. This pattern in fact applies for any real values of $n$, so it can be used also for computing derivatives for $f_3(x)=\sqrt{x}$ above and $g_3(x)=1/x$.