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

Key Equations

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

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

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

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 [latex]f(x)[/latex] at [latex]a[/latex] is given by

[latex]\dfrac{f(a+h)-f(a)}{h}[/latex] or [latex]\dfrac{f(x)-f(a)}{x-a}[/latex]

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