Derivative Basics
Simply saying, it’s just the SLOPE of ONE POINT of a graph (line or curves or anything).
Refer to Mathsisfun: Introduction to Derivatives
A Derivative, is the Instantaneous Rate of Change
, which's related to the tangent line
of a point, instead of a secant line to calculate the Average rate of change.
“Derivatives are the result of performing a differentiation process upon a function or an expression. ”
Derivative notations
Refer to Khan academy article: Derivative notation review.
Lagrange's notation
In Lagrange’s notation, the derivative of f(x)
expressed as f'(x)
, reads as f prime of x
.
Leibniz's notation
In this form, we write dx
instead of Δx heads towards 0
.
And the derivative of
is commonly written as:
For memorizing, just see
d
asΔ
, readsDelta
, means change. Sody/dx
meansΔy/Δx
. Or it can be represent asdf / dx
ord/dx · f(x)
, whatever.
How to understand dy/dx
This is a review from “the future”, which means while studying Calculus, you have to come back constantly to review what the dy/dx
means. ---- It's just so confusing.
Without fully understanding the dy/dx
, you will be lost at topics like Differentiate Implicit functions
, Related Rates
, Differential Equations
and such.
Tangent line & Secant line
- The
secant line
is drawn to connect TWO POINTS, and gets us theAverage Rate of Change
between two points. - The
Tangent line
is drawn through ONE POINT, and gets us theRage of change at the exact moment
.
As for the secant line
, its interval gets smaller and smaller and APPROACHING to 0
distance, it actually is a process of calculating limits
approaching 0
, which will get us the tangent line
, that been said, is the whole business we're talking about: the Derivative
, the Instantaneous Rate of Change
.
Secant line
Example
Solve:
- What it’s asking is the
Slope of its secant line
:
- which could be applied with this simple formula:
- the result is
( f(3)-f(1) )/ (3-1) = -1/12