apcalcDifferentiation+Rules

**__The derivative of a function__** f at any number denoted by f’(x) is given by: f’(x)=
 * Differentiation Rules ﻿ **
 * A derivative can be described as the either the slope of a tangent line on a curve at any point or as the rate of change
 * A function is differentiable at =a if f’(a) exists.
 * A function is differentiable on (a, b) if it is differentiable for all on that interval.
 * A function is not differentiable if:
 * The graph has a cusp


 * The graph has a corner




 * A function has a discontinuity: a hole, infinite (as shown in picture)




 * Where a function has a vertical tangent

** Notation **


 * The derivative of f(x) can be notated as f’(x)
 * ** Df ** and **Dxf**
 * If **y = f(x)**, then **y'** can be used to denote the derivative of the function **f**.
 * If we let **Dx = h** and **Dy = f(x + h) - f(x)** then



can denote the derivative of the function **f**. Find the derivative of f(x) =-7x+6 at any point x.
 * Example: **



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(The derivative of the outside equation multiplied by the derivative of the inside equation)

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STILL confused? Check out this video: [] **__ History of Differentiation Rules __** Gottfried [|Leibniz]and Isaac **[|Newton] **. are considered the “fathers of calculus”. Newton specifically was the first to introduce the notion of higher derivatives, the chain rule, and the product rule in order to assist him in solving mathematical physics problems. However, Newton never officially published all of his findings because his findings at the time were considered disreputable.

Leibniz was the first man to take these ideas and form them into “true calculus”. He was originally accused of plagiarizing Newton, however he is now considered as an independent contributor and inventor of calculus. Leibniz spent a great deal of time creating the correct forms and notation for different differential equations, specifically the chain and product rules, in their integral and differential forms. His new formations allowed for the computations of second and higher derivatives.

Today both Newton and Leibniz are both credited with contributing to the creation of calculus as we know it. Newton was the first to apply the findings to physics while Leibniz developed a great deal of the notation used today in calculus. Basically, Leibniz and Newton were the founding fathers of the laws of differentiation and integration, second and higher derivatives, and notation. ([])

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