50DiffTheory

Theory of Differentiation

For any function, the **average rate of change** between two points is the same as the **__gradient__ of a straight line** between those two points.

math \text{Gradient } = \dfrac {\Delta y}{\Delta x} = \dfrac {y_2 - y_1}{x_2 - x_1} math Note that D (delta) is the Greek symbol for capital D and is used to denote "change."

ie D x is read as "change in x."



Using function notation, and using D x = h, this gradient rule becomes: math \dfrac {\Delta f}{\Delta x} = \dfrac {f(x+h) - f(x)}{h} math


 * Differentiation by First Principles **

The **instantaneous rate of change at a point** (otherwise known as the **derivative** ) is defined by the limit of this gradient as h --> 0 math f'(x) = \begin{matrix} \lim \\ h \to 0 \\ \end{matrix} \; \dfrac {f(x+h) - f(x)}{h} math This gradient function is usually referred to as the **derivative by first principles**.

The instantantaneous rate of change at a point also represents the gradient of the tangent to the curve at that point.

The alternative notation for derivative is: math \dfrac {dy}{dx} \text{ or sometimes } \dfrac {d}{dx}(y) math

In the derivative, **dx** can be read as "//**an infinitesimal change in x**//"

Note: The derivative is only defined if the function is __continuous__ and __smooth__. This means there is no derivative at an endpoint or at a "pointy bit."



f '(x) is undefined at x = 1 and at x = 3

{Because of the "pointy bits" the function is not smooth at those two points}



g'(x) is undefined at x = 1 {Because of the endpoint}

g'(x) is undefined for x < 1 {Because g(x) is undefined for those values}

g'(x) is undefined at x = 3.5 {Because of the hole (discontinuity) }

**See also** [|www.mathsonline.com.au] Year12Advanced --> Calculus --> Introductory Calculus --> Lesson 2

Historical Note The traditional way to write the gradient function was to use d // x //instead of **h**.

d (delta) is the Greek symbol for lower case "d" and is used to denote "a small change."

ie d // x //is read as "a small change in x."

The gradient function was therefore: math f'(x) = \begin{matrix} \lim \\ \delta x \to 0 \\ \end{matrix} \; \dfrac {f(x+\delta x) - f(x)}{\delta x} math or math \dfrac {dy}{dx} = \begin{matrix} \lim \\ \delta x \to 0 \\ \end{matrix} \; \dfrac {\delta y}{\delta x} math Notice the __deliberate__ transformation from D x to d x to dx.

math \dfrac {\Delta y}{\Delta x} \Longrightarrow \dfrac {\delta y}{\delta x} \Longrightarrow \dfrac{dy}{dx} math

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