The Derivative

Mathematics is the science that is very widespread is also fun, previously we have discussed about the frequency of the hope and the opportunity tocomplement an event which is a sub topic of discussion opportunities. 
And this time the topic we will learn about the derivative? You must have to know what'sderivative is not it?

A derivative is a measurement of how a function changes over against changesthe value of the inputs, or in general the derivative showed how one quantity is changing due to changes of other quantities. The process of finding the derivative is called differentiation.


  • 02
  • 03
  • 04
  • 05

  • 06  is a symbol for the first derivative.
  • 06 is  a symbol for the second derivative.
  • 06 is a symbol for the third derivative.
    other symbols other than the 06
  •  and 06
  •  is 07
  •  and 08

FIRST DERIVATIVE

Suppose y is a function of x or can be written also y = f (x). The derivative of yagainst x dinotasikan as follows:



09




Used with the above definition of the derivative can be lowered severalderivative formulas, namely:


1. If known  010 dimana C and n real constants, then the 011

Consider the following example:

012


2. If known y = C and    
013

Consider the following example:
014

3. for
015

:
016
4. for y = f (x) g (x) then 
017
or
example:
018


5.
021
6. For other derivatives are present in the explanation below.
001


SECOND DERIVATIVE


The second derivative of y = f (x) of x dinotasikan as follows
023
The second derivative is the derivative obtained by lowering the back of the firstderivative. Consider the following example:
002

The use for the second derivative is among other things to:


a. determine the gradient of the tangent line of the curve

If known to the offending line of g of the curve y = f (x) on the point (a, f (a)) so that the gradient for g is
024


For example specify the gradient of the tangent line of the curve y = x ² + 3 x emphasis (1.0-4)!


Solution:
025


So the gradient of the tangent line of the curve y = x ² + 3 x emphasis (1.0-4) is m = y (1) = 2.1 + 3 = 5


b. determine if the intervals up or down

the curve y = f (x) ride if f ' (x) > 0 and the curve y = f (x) down if f ' (x) < 0. thenhow to determine f ' (x) > 0 or f ' (x) < 0? we use line numbers of f ' (x). Consider the following example:

Specify the interval interval going up and down from the function y = x ³ + 3 x²-24 x!

Answer:

y = f (x) = x ³ + 3 x ²-24 x  f ' (x) = 3 x ² + 6 x-24 = 3 (x ² + 2 ×-8) = 3 (x + 3) (x-2)

026

Based on line numbers obtained above:

f ' (x) > 0 for x <-4 and x > 2 which is the interval for the up.

F ' (x) 4-0 for < < x 2 which is the < interval for the function down.

c. specify the maximum value and minimum value

The maximum value and the minimum value of the function is often called also with extreme value or the value of the stationary function, which can be obtained on f ' (x) = 0 for the function y = f (x). For more details look at the following example.

Specify the value of the extreme of a function y = x ²-3 x ³-24 × 7!

Answer:

y ' = 3 x ²-6 x-24

extreme values retrieved from y ' = o then

3 x ²-6 x-24 = 0

(x ²-2 x-8) = 0

(x-4) (x + 2) = 0

x 1 = 4; x 2 =-2

027

Based on line numbers above:

The maximum of the function at x =-2 so that the value of the maximumfeedback are:

f (-2) = (-2) ³-3 (-2) ²-24 (-2)-7

f (-2) = 21

The minimum of the function at x = 4 so the value behind the minimum of:

f (4) = (4) ³-3 (4) ²-24 (4)-7

f (4) =-87



DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

The following formula for the derivative of trigonometric functions:
028
029
030
Consider the following example:
031
Answer:
032
033
What is the explanation of the derivation of the above have made you reallyunderstand about derivatives and have been able to work on a variety of aderivative that will be a matter of variation you encounter. Let's hope it is so. Asenter thou learning problems so that you better understand any materials of mathematics. For it in learning so what can aspired was achieved, see also articleOpportunities Compound Events and Conditional Events from sub chapter topicopportunities.

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