▶️Differential Calculus

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

For memorizing, just see d as Δ, reads Delta, means change. So dy/dx means Δy/Δx. Or it can be represent as df / dx or d/dx · f(x), whatever.

How to understand dy/dx

Refer to Khan academy from Differential Equation section: Addressing treating differentials algebraically

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 the Average Rate of Change between two points.

  • The Tangent line is drawn through ONE POINT, and gets us the Rage 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

  • 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

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