07Cderivative

toc = The derivative of //x// n =

If .. f(x) = x n
 * then .. f ' (x) = nx n –1


 * ** bring the index down and multiply it at the front of the term **
 * ** subtract 1 from the index **

Eg if .. f(x) = x 4
 * then .. f '(x) = 4x 3.

If .. f(x) = ax n .... // {where a is any constant} //
 * then .. f ' (x) = anx n – 1

Eg if .. f(x) = 5x 4 ,
 * then .. f '(x) = 20x 3

If .. f (x) = a ..... // {where a is any constant} //
 * then .. f ' (x) = 0 .... { // because a = ax0 //, so 0 × a = 0}

Eg if .. f(x) = 5,
 * then .. f '(x) = 0


 * The derivative of Polynomials **


 * To find the derivative of any polynomial
 * find the derivative of each term and add the results


 * Example **

... ... Find the derivative of .. f(x) = 3x 2 – 7x + 3


 * Solution:**

math . \qquad f(x) = 3x^2 - 7x^1 + 3x^0 \qquad. \\.\\ . \qquad f '(x) = 2 \big( 3x^{2-1} \big) - 1 \big( 7x^{1-1} \big) + 0 \big( x^{0-1} \big) \qquad. \\.\\ . \qquad f '(x) = 6x^1 - 7x^0 + 0 \qquad. \\.\\ . \qquad f '(x) = 6x - 7 math


 * Note 1 **

math . \qquad \text{Remember that } \; \dfrac{dy}{dx} \; \text{ is an alternate notation for derivative} \qquad. \\.\\.\\ . \qquad \text{if } \; y = x^2 + 3x + 4 \\.\\ . \qquad \qquad \dfrac{dy}{dx} = 2x + 3 math


 * Note 2 **
 * These rules work even if n is negative or a fraction
 * ** bring the index down and multiply it at the front of the term **
 * ** subtract one from the index **

math . \qquad \text{Recall that} : \\.\\ . \qquad \qquad \dfrac{1}{x^3} = x^{-3} \qquad. \\.\\ . \\ . \qquad \text{if } \; y = x^{-3} \\.\\ . \qquad \qquad \dfrac{dy}{dx} = -3x^{-4} \qquad. \\.\\ math . math . \qquad \text{Recall that} : \\.\\ . \qquad \qquad \sqrt {x} = x^{\dfrac{1}{2}} \quad and \quad \sqrt [3] {x} = x^{\dfrac{1}{3}} \qquad. \\.\\.\\ . \qquad \text{if } \; y = x^{\dfrac{1}{3}} \\.\\ . \qquad \qquad \dfrac{dy}{dx} = \dfrac{1}{3}x^{-\dfrac{2}{3}} math

Differentiation on the Calculator



 * Your calculator can do differentiation.


 * Go to the ** 2D ** tab of the virtual keyboard.
 * Select ** CALC ** from the bottom row and select the differentiation form (circled in red).


 * You will need to put // **x** // into the denominator of the derivative.


 * Then enter the expression to be differentiated between the brackets and press EXE.


 * Avoid writing calculator notation when you answer questions.
 * Eg write 2x and not 2 • x

**Example 1**


... ... Find the derivative of .. //y = x 2 – 5x// – //24//


 * Solution:**

math \\ . \qquad y=x^2-5x-24 \qquad. \\.\\.\\ . \qquad \dfrac{dy}{dx}=2x-5 \qquad. math

** Example 2 **
math \\ . \qquad \text{Find the derivative of } \; f(x)=\dfrac{3x^2-2x+1}{x} \qquad. \\ . \\ . \qquad \text{ and hence find } f '(1) math


 * Solution:**

math \\ . \qquad f(x) = \dfrac{3x^2-2x+1}{x} \qquad. \\ . \\ \big\{ \text{break into separate fractions} \big\} \qquad. \\ . \\ . \qquad f(x) = \dfrac{3x^2}{x} - \dfrac{2x}{x} + \dfrac{1}{x} \qquad. \\.\\ . \qquad f(x) =3x-2+\dfrac{1}{x} \\. \\ .\qquad f(x) = 3x-2+x^{-1} math


 * ** therefore the derivative is: **

math \\ . \qquad f ' (x)=3-0-x^{-2} \qquad. \\ . \\ . \qquad f '(x) = 3-\dfrac{1}{x^2} \qquad. math


 * ** and hence find f '(1) **

math \\ . \qquad f ' (1)=3-\dfrac{1}{1^2} \qquad. \\ . \\ . \qquad f '(1) =2 math


 * Note 1 **
 * On your calculator, entering "| x = 1" after the derivative gives f '(1).
 * The "|" symbol is in the ** OPTN ** section of either the ** MTH ** tab or the ** 2D ** tab.


 * Note 2 **
 * The calculator will sometimes give the result in a different format from the way you obtain your answer
 * see answer for Example 2 on right


 * The expand command will split the combined fraction into separate fractions
 * (from the ACTION menu, TRANSFORMATION submenu)
 * The simplify command (same place) will usually do the same thing.
 * The combine command (same place) will combine 2 fractions into a single fraction.


 * Note 3 **
 * If you intend to do several calculations with the same function, it is worth defining the function, first.
 * The define command is in the ACTION menu, COMMAND submenu.
 * Use "f" from the ABC tab of the virtual keyboard, but don't use "x" from the same place. Use the variable version of "x".

**Example 3**


math . \qquad \text{Differentiate } \;y=\sqrt{x}-\dfrac{2}{\sqrt{x}} \qquad. math


 * Solution:**

math \\ . \qquad y = \sqrt{x} - \dfrac{2}{\sqrt{x}}\qquad. \\ . \\ . \qquad y = x^{\dfrac{1}{2}} - 2x^{-\dfrac{1}{2}} \qquad. \\.\\ math

math \\ . \qquad \dfrac{dy}{dx} = \dfrac{1}{2} x^ {-\dfrac{1}{2}}-2 \left( -\dfrac{1}{2}x^{-\dfrac{3}{2}} \right) \qquad. \\ . \\ . \qquad \dfrac{dy}{dx} = \dfrac{1}{2\sqrt{x}}+\dfrac{1}{\sqrt{x^3}} \qquad. math

Note: Go to top of page flat
 * A function can only be differentiated at a point if its graph is __**continuous**__ and __**smooth**__ at that point.
 * Also, we can not differentiate the __**endpoint**__ of a function.
 * This is because differentiating is equivalent to finding a tangent to the curve at that point
 * If the graph is discontinuous, or an endpoint, or not smooth then the tangent is undefined.
 * See here for the conditions of differentiation.

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