Summary of Defining the Derivative

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 $f(x)$ 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(t) at time t is the derivative of the position s(t) at time t.
- Average velocity is given by
  $v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}$
- Instantaneous velocity is given by
  
  $v(a)=s^{\prime}(a)=\underset{t\to a}{\lim}\dfrac{s(t)-s(a)}{t-a}$
We may estimate a derivative by using a table of values.

Key Equations

Difference quotient
$Q=\dfrac{f(x)-f(a)}{x-a}$
Difference quotient with increment $h$
$Q=\dfrac{f(a+h)-f(a)}{a+h-a}=\dfrac{f(a+h)-f(a)}{h}$
Slope of tangent line
$m_{\tan}=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}$

$m_{\tan}=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}$
Derivative of $f(x)$ at $a$
$f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}$

$f^{\prime}(a)=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}$
Average velocity
$v_{\text{avg}}=\dfrac{s(t)-s(a)}{t-a}$
Instantaneous velocity
$v(a)=s^{\prime}(a)=\underset{t\to a}{\lim}\dfrac{s(t)-s(a)}{t-a}$

derivative: the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative

difference quotient: of a function $f(x)$ at $a$ is given by

$\dfrac{f(a+h)-f(a)}{h}$ or $\dfrac{f(x)-f(a)}{x-a}$

instantaneous rate of change: the rate of change of a function at any point along the function $a$ , also called $f^{\prime}(a)$ , or the derivative of the function at $a$