3.1 Defining the derivative  (Page 5/10)

Key concepts

• The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment $h.$
• The derivative of a function $f\left(x\right)$ at a value $a$ is found using either of the definitions for the slope of the tangent line.
• Velocity is the rate of change of position. As such, the velocity $v\left(t\right)$ at time $t$ is the derivative of the position $s\left(t\right)$ at time $t.$ Average velocity is given by
  
  $v_{\text{ave}}=\frac{s\left(t\right)-s(a)}{t-a}.$
  
  Instantaneous velocity is given by
  
  $v\left(a\right)=s^{\prime }\left(a\right)=\underset{t\to a}{\text{lim}}\frac{s\left(t\right)-s(a)}{t-a}.$
• We may estimate a derivative by using a table of values.

Key equations

• Difference quotient
  $Q=\frac{f\left(x\right)-f(a)}{x-a}$
• Difference quotient with increment $h$
  $Q=\frac{f\left(a+h\right)-f(a)}{a+h-a}=\frac{f\left(a+h\right)-f(a)}{h}$
• Slope of tangent line
  $m_{\text{tan}}=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f(a)}{x-a}$
  $m_{\text{tan}}=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f(a)}{h}$
• Derivative of $f(x)$ at $a$
  $f^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$
  $f^{\prime }(a)=\underset{h\to 0}{\text{lim}}\frac{f\left(a+h\right)-f(a)}{h}$
• Average velocity
  $v_{a\text{ve}}=\frac{s\left(t\right)-s(a)}{t-a}$
• Instantaneous velocity
  $v\left(a\right)=s^{\prime }\left(a\right)=\underset{t\to a}{\text{lim}}\frac{s\left(t\right)-s(a)}{t-a}$

For the following exercises, use [link] to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f\left(x\right).$