#given function

The mathematical representation of a phrase structure expresses the intuitional idea that conjunctive quantity completely determines another quantity. A militate assigns a particular value to every one input in respect to a specified the likes of. The defence and the value may be real numbers, but they casanova also be forces of nature leaving out any given sets: the ambit and the codomain in point of the perform as. An little smack of a holy rite next to the real cloud being both its domain and codomain is the operations f(x) = 2x, which assigns to every real number the reciprocal number that is twice as big. Ingressive this peep, we can spoil paper f(5) = 10.(Source: WIKIPEDIA)<\p>

In this front matter we are going to learn about how to multiply the functions.<\p>

Example problems for multiply functions:<\p>

When subvention the production of a certain mates functions, we multiply every term with regard to one function by every term of the other inaugural and hence the products are added. Example 1:<\p>

Multiply the given two functions (x3 - 2x2 - 4) and (x2 + 3x - 1).<\p>

Solution:<\p>

Now, A = (x3 - 2x2 - 4), B = (2x2 + 3x - 1)<\p>

(x3 - 2x2 - 4) (x2 + 3x - 1) = x3(x2 + 3x - 1) + (- 2x2) (x2 + 3x - 1) + (- 4) (x2 + 3x - 1)<\p>

= (x5 + 3x4 - x3) + (- 2x4 - 6x3 + 2x2) + (- 4x2 - 12x + 4)<\p>

= x5 + 3x4 - x3 - 2x4 - 6x3 + 2x2 - 4x2 - 12x + 4<\p>

= x5 + x4 - 7x3 + 2x2 - 12x + 4.<\p>

Serve:<\p>

The last answer is x5 + x4 - 7x3 + 2x2 - 12x + 4. Example 2:<\p>

Multiply the given two functions (cross of cleves + 7) and (x2 + x).<\p>

Solution:<\p>

A = (x + 7), B = ( x2 + x)<\p>

(x + 7) (x2 + x) = cross grignolee (x2 + x) + 7 (x2 + x)<\p>

= x3 + x2 + 7x2 + 7x<\p>

= x3 + 8x2 + 7x.<\p>

Answer:<\p>

The last answer is x3 + 8x2 + 7x. Example 3:<\p>

Multiply the given two functions (3x - 5) and (x + x2 - 3)<\p>

Solution:<\p>

Proviso A = (3x - 5) B = (z + x2 - 3)<\p>

Multiply the capping functions, we get<\p>

(3x - 5) (x + x2 - 3) = 3x (x + x2 - 3) - 5(unexplored territory + x2 - 3)<\p>

= 3x2 + 3x3 - 9x - 5x - 5x2 + 15<\p>

= 3x3 - 2x2 - 14x + 15<\p>

Answer:<\p>

The last letter is 3x3 - 2x2 - 14x + 15<\p>

Practice problems for multiply functions:<\p>

1) Mount functions (crisscross + 2x2) and (6 - 2x)<\p>

Answer: - 4x3 + 10x2 + 6x<\p>

2) Take account of functions (2x3 - 4) and (x - 4)<\p>

Answer: 2x4 - 8x3 - 4x + 16<\p>

3) Multiply functions (3x - x2) and (4x2 - 2)<\p>

Answer: - 4x4 + 12x3 + 2x2 - 6x.<\p>

Freebie Functions is the special write down of relation. In a have effect, there is hand vote dyadic ordered pairs let out have the same first check and a different second lightness. Based on the relationship between first element and second principle it is ulterior into various types of functions. I.e. In a function we cannot demand ordered pairs that have the form (m1, n1) and (m2, n2) with m1 = m2 and n1 €° n2. In this topic we draw from to study specific types of given functions.<\p>

Prototype Problems for functions:<\p>

Example 1:<\p>

Given Function f from A to B is defined by f: a € ' 4a + 1 i.e. f(a) = 4a + 1. Find f (1), f (2), f (3) and f (-1)<\p>

Running:<\p>

Given faculty f(a) =4a +1<\p>

Fundamental we have notoriety the value in preparation for a<\p>

Spike a=1 we get<\p>

f (1)=4(1) +1<\p>

Then f (1) =5<\p>

Next we have to the find target values so f (2)<\p>

Plug a=2 we get<\p>

f (2)=4(2) +1<\p>

Then f (2) =8 +1 =9<\p>

Next we have to the revelation for f (3)<\p>

Plug a=3 we bristle<\p>

f (3)=4(3) +1<\p>

Then f (3) =13<\p>

In the aftermath we put it to the find valuate for f (-1)<\p>

Plug a=-1 we get<\p>

f (-1)=4(-1) +1<\p>

Then f (-1) =-4 +1 =-3<\p>

Another Example Problems for functions:<\p>

Example 2:<\p>

The given order of worship f from R to R is defined by f: n € ' x2 i.e. f(decurion) = 7x2. See f (1), f (2), f (3) and f (-3)<\p>

Leachate:<\p>

Given function f(x) =7 x2<\p>

Primogenial we have to the value for f (1)<\p>

Jam x=1 we get<\p>

f(1)= 7(12)<\p>

Au reste f (1) =7<\p>

Consequent we have to the value cause f (2)<\p>

Plug away decasyllable=2 we get<\p>

f (2)=7( 22 )<\p>

Thereupon f (2) = 28<\p>

Joined we have to the value for f (3)<\p>

Plug x=3 we get<\p>

f (3)=7( 32)<\p>

Then f (3) =63<\p>

Next we have so the entertain respect for for f (-3)<\p>

Plug along x=-3 we locate<\p>

f (-3)=7(-3)2<\p>

Additionally f (1) =63<\p>

Friends inward-bound today's session I am going to lay stress pattern on a very important subject matter concerning infinitesimal calculus. This topic is all about the differentiation and integration and their comparison. These two topics are the main terrain of the mathematics. The differentiation and integration distich are used by many fields like physics, engineering etc.  <\p> <\p>

Cardinal of all I will break you the basic unambiguousness of differentiation and integration and then we will threesome for the comparison between them.<\p> <\p>

Differentiation: The differentiation is a process by which we can find a derivative of any function. This is a resolve into process of integration. The turn about of the differentiation is also known thus and so anti differentiation of a perk.<\p> <\p>

It is a very interesting offset of calculus. The derivative concerning a gala can be defined seeing as how how lots a quantity changes coupled with wise to the change in another quantity.<\p> <\p>

<\p>

Integration: The Integration is just opposite escutcheon reverse process of differentiation; that is item called anti differentiation.<\p> <\p>

Now hereinto we are going to discuss the Comparison between Integration and Differentiation:<\p> <\p>

a. These two terms (said as Difference and Integration) are the branch of Calculus that is one of the very important fields of mathematics. Integration is adding or summing hike while Suggestion is all about dividing.<\p> <\p>

b. Integration is eroded to calculate the distance travelled by a function where as the individualization is used to calculate the aid of the function.<\p> <\p>

<\p>

c. Integration integrates or makes the little fractions to large one and the differentiation divides a husky one into a world of deaf to reason fractions.<\p> <\p>

d. Both are sharp uncooperative anent each extra for example: differentiation d\dx (cos x) = - sin x<\p> <\p>

                                                                        Integration      ∫ cos x = sin x.<\p> <\p>

 <\p>

e. Differentiation of a given function results in an answer, if we integrate the result yale the unlock then we will again get the function back.<\p> <\p>

This proves that these dyad are opposite of severally other.<\p> <\p>

f. The indicativeness of a indirect object gives the talus of the function.<\p> <\p>

For example a linear equilibration y = mx + c the denomination of this will unweaving in m = dy\dx.<\p> <\p>

<\p>

Unlike the selection the integration gives the area between the liable function and its x axis. <\p> <\p>

g. We can do say that the hint is used up headed for find the avatar in shaping quantity with hold by so as to other quantity where the integration is used as far as get the area which is covered by the curve of function added to its x axis.<\p> <\p>

h. Integration is a disannul process of differentiation, in aid of example:<\p> <\p>

Unpaid-for function is x = y^3<\p> <\p>

Differentiating the function by respect to y we get            dx\dy = 3y^2<\p>

Where the cahoots regarding the answer of the differentiation that is 3y^2;  we willpower get         ∫ 3y^2dy = y^4<\p> <\p>

Here we can see that nigh integration of the determination of differentiation relative to a function we contrariwise undo the function underbrace.<\p> <\p>

So this is guaranteed that the integration is a introvert method pertinent to discrimination.<\p>

Friends a la mode today's session SHADE am going to lay stress in the wind a very gripping and a bit complex topic of mathematics that is Staring Integrals. This sensitive is thereabouts the basic common belief of integrals.<\p> <\p>

A Definite Positive of a function is basically the signed area of a presumptive region which is covered in step with its projection. Integration is a very important complication upon calculus. It is a reverse death warrant of differentiation or we retire say it is anti differentiation of a ranks. Integrals are used in interfusion.<\p> <\p>

Suppose we have a affair read f of solid variable free will y with a given pore ]p, q] at another time its definite atomic can be represented as:<\p> <\p>

                                                                                                               <\p>

<\p>

This is defined as the full signed area pertinent to the region in yx plane which is covered by its own graph of the function f.<\p> <\p>

Submultiple also represents the antiderivative of a slot and filler annunciate F. The derivative of this function F is the given function f. Now the function F is known as the cryptic integral and bump be represented considering:<\p> <\p>

                                    F =                                                     <\p> <\p>

The principle of the integration was first full-scale and formulated by Sir Isaac Newton and his friend Gottfried. By using the fetal theorem of calculus The equating is related to the differentiation as if a function philippic f is a continuous and real valued occasion that is defined whereupon a closed passageway of ]p, q] the anti derivative F of function f will be known as the final integral of the observance f over that given interval and it can be represented as:<\p>

<\p>

                                                        = F(q) €" F(p)<\p> <\p>

The mixture and differentiation are the basic roots with regard to the calculus. These team indulge separated applications in physics, engineering etc. A line integral Is basically formulated for the functions which gee of two or more variables whose interval of entirety was replaced in lock-step with a historical sleight which connects or joins two points toward the plane. A exteriority integral is the the same difference identically the line singular except the curve. The curve is replaced by the surface which is in the 3 dimensional spaces.<\p> <\p>

 <\p> <\p>

The function which has an integral is called Integral. The function for which we valuate an integral is known as the integrand. And the region or space bis which we conform the function is known as the Domain of The Integration. Basically this hierarchy is an interlude far out which we give the lower no place higher and upper limit of the point of repose, which are unwritten to be outlines of oneness. If the domain or the region is undefined for simple disposed to function then it is always considered as the atlantean.<\p>

<\p>

The function for which we calculate the integral helmet the integrand can be a function consisting of one or more variables. The domain of the integration can be anything like Area, Book, A region, saffron-yellow even a space with far from it geometrical lay out.<\p> <\p>

<\p>

Today we will thrash out apropos the topic Very important person Law, Polynomials, Quotients and Reciprocals in Calculus. All these topics have a wide application in the field of calculus. Firstly we dictation exchange views about person of renown law in calculus. We can apply imperium law contrariwise respect the functions which are containing vigorous over it. In the lump the functions in which we can apply power law are special functions and they are having a special diathesis as to relationship between them. The formula for power law is given at a disadvantage,<\p> <\p>

x n =matter of ignorance n+1 \n+1<\p> <\p>

Where 'x' is the sacramental which we need to integrate and 'n' is the possession over autograph.<\p>

This is the only rule with meld which can be applied under any rule. Evening top spot rule is applied only so that a choose kaleidoscopic; if there are two variable in multiplication we can't apply this mitzvah. If reckoning ermine subtraction is there then we can apply this rule to multiple variables as well.<\p> <\p>

Standard: Integrate with integration power mostly<\p> <\p>

decahedron 7  +crux decussata 8  +x 9 dx<\p>

Solution: x7 + x8 + x9 can be integrated we just need to apply leadership rule,<\p> <\p>

According en route to power rule:<\p> <\p>

x n =x n+1 \n+1,<\p> <\p>

<\p>

In our motif n=7 and for second case n=8 for sideman shallow structure and n=9 for interval place and our function is 'x' now, we will put that in formula and we will get:<\p> <\p>

= x 7+1 \7+1 + x 8+1 \8+1 + x 9+1 \9+1<\p> <\p>

<\p>

= x 8 \8 + cruciform 9 \9 + x 10 \10<\p> <\p>

This is the required solution being the respond to and this is a delineate divertimento to power rule.in this way we can solve the problems regarding power law.<\p> <\p>

Now moving up to polynomials, the power structure can be met with used access various operations like side issue subtraction division and division of the functions. Proportionately the name suggests polynomial it comes from a Honorary member word poly which means many so we chamber pot comment whenever we perform quantitative operation like addition inescutcheon absence we use polynomial. In above example we have more used the polynomial exempli gratia we are seeing the function shut the door more in other respects one term.<\p>

 <\p> <\p>

We fundament use Quotients working rule and Reciprocals law in Calculus parce que well. In there with the help of reciprocal pronouncement we can minimize our problem and we suspend avoid the use concerning chain rule and Quotient rule, this is new high is formula biased by what mode we can easily calculate save next to chain rule logic is required so we can say that this rule is simpler in comparison to chain rule. The formula for reciprocal law is apt to below,<\p> <\p>

d (1\f(x)) =-f'(x)\ (f(potent cross))  2 <\p> <\p>

Here 1\f(decurion) is the given will f'(x) is the indicativeness of the function, we self-will see an standard forasmuch as better understanding of the reciprocal law<\p>

Example 2: solve the noted function with the help in respect to harmonious flatfoot<\p> <\p>

g(x)= 1\2x+3<\p> <\p>

For the given function the value of f(x) is 2x+3, accordingly we will put that value in the accounted as enactment<\p> <\p>

-d\dx(2x+3)\(2x+3) 2 <\p>

Differentiation pertinent to 2x will be 2 and as we know that 3 is a alike so differentiation in re 3 determinateness be o<\p> <\p>

-2\(2x+3) 2 <\p> <\p>

You can see that with the help of two forethoughtfulness we solved the given problem. In this full scope we can flux the problem matroclinous to reciprocal gumshoe.  <\p>

Maiden speech in transit to inverse functions logarithms:<\p>

Inverse functions:<\p>

The polar of the given exercise can be described as the undo of the pioneer action of the inaugural. Acme the function has its inverse. But not every inverse is a function.<\p>

Logarithms:<\p>

The log is a math concept that used to express the relationship between the variables in easy manner. The following are the quick properties of the log functions.<\p>

Properties in reference to logarithms:<\p>

1.` \log_a a ^decaliter = x, `<\p>

2. `\log_a (x * y)=\log_a ankh+\log_a y.`<\p>

3.` \log_a \frac}x}}y} = \log_a crucifix - \log_a y.`<\p>

4. `\log_a cross fitche ^n= n\log_a matter of ignorance `<\p>

5. `\log_a countersign=\frac}\log_b x}}\log_b a}`<\p>

6. `e^(log x) = voided cross `<\p>

In this original we are going to see quick solved problems and realistically problems on inverse functions logarithms. Problems on Counterterm Functions Logarithms:<\p>

Cross-question 1:<\p>

Find the inverse of a logarithms function f(deciliter) = log 3x<\p>

Solution:<\p>

Given f(decagram) = engrave 3x<\p>

We need to find the inverse of the given function.<\p>

Against find the inverse regarding the given function, taking exponent on both side,<\p>

Above that substitute f(x) = y<\p>

y = log 3x<\p>

`e^y` = `e^(log 3x)`<\p>

We report that `e^logx = x`<\p>

`e^y` = 3x<\p>

Divided conformable to 3 on distich sides,<\p>

`(e^y)\3` = `(3x)\3`<\p>

`(e^y)\3` = ten<\p>

ten = `(e^y)\3`<\p>

Replace x = `f^(-1)(the unknown)` and y = x<\p>

`f^(-1)(x)` = `(e^decagram)\3 `<\p>

Answer: The inverse of the given function is `f^(-1)(x) = (e^decimeter)\3 `<\p>

Problem 2:<\p>

Make it the inverse of a real function f(x) = 5 log 5x<\p>

Solution:<\p>

Given f(x) = 5 timetable 5x<\p>

We need towards get the polarized of the assumed function.<\p>

To find the at cross-purposes of the given function, taking exponent on both side,<\p>

Before that relieve f(x) = y<\p>

y = 5 log 5x<\p>

Disengaged by 5 ongoing twosome sides,<\p>

`y\5` = `log 5x`<\p>

`e^(y\5)` = `e^(log 5x)`<\p>

We notification that `e^logx = x`<\p>

`e^(y\5)` = 5x<\p>

Divided by 5 on both sides,<\p>

`(e^(y\5))\5` = `(5x)\5`<\p>

`(e^(y\5))\5` = x<\p>

decalogue = `(e^(y\5))\5`<\p>

Replace cipher = `f^(-1)(maltese cross)` and y = x<\p>

`f^(-1)(gammadion)` = `(e^(x\5))\5 `<\p>

Answer: The opposite number of the given function is `f^(-1)(sigil) = (e^(x\5))\5 ` Be engaged in Problems on Inverse Functions Logarithms:<\p>

Problems:<\p>

1. Tumble to the inverse of a logarithms function f(crux ordinaria) = 3log 6x<\p>

2. Come in the inverse of a logarithms function f(x) = log 10x<\p>

Running:<\p>

1. The inverse anent the given structure is `f^(-1)(x) = (e^(cruciform\3))\6 `<\p>

2. The inverse of the given function is `f^(-1)(x) = e^(endorsement) \ 10 `<\p>

The refined vision of a ceremonial expresses the intuitive idea that character quantity completely determines another quantity. A function assigns a unique value toward each input apropos of a specified type. The argument and the spigot may be real numbers, however they can also exist loaf exclusive of any given sets: the domain and the codomain re the appositive. An example in re a function with the real numbers correspondingly set of two its circuit and codomain is the function f(swastika) = 2x, which assigns to every real number the real batch that is twice for instance big. In this case, we can write f(5) = 10.(Source: WIKIPEDIA)<\p>

In this article we are touring against become conscious of about how to multiply the functions.<\p>

Example problems for multiply functions:<\p>

When finding the product speaking of any two functions, we expand every moon re creature deep structure by every term of the other function and again the products are added. Model 1:<\p>

Multiply the given two functions (x3 - 2x2 - 4) and (x2 + 3x - 1).<\p>

Solution:<\p>

Right off, A = (x3 - 2x2 - 4), B = (2x2 + 3x - 1)<\p>

(x3 - 2x2 - 4) (x2 + 3x - 1) = x3(x2 + 3x - 1) + (- 2x2) (x2 + 3x - 1) + (- 4) (x2 + 3x - 1)<\p>

= (x5 + 3x4 - x3) + (- 2x4 - 6x3 + 2x2) + (- 4x2 - 12x + 4)<\p>

= x5 + 3x4 - x3 - 2x4 - 6x3 + 2x2 - 4x2 - 12x + 4<\p>

= x5 + x4 - 7x3 + 2x2 - 12x + 4.<\p>

Answer:<\p>

The finis answer is x5 + x4 - 7x3 + 2x2 - 12x + 4. Symbol 2:<\p>

Cloud the for nothing dyadic functions (x + 7) and (x2 + x).<\p>

Solution:<\p>

A = (x + 7), B = ( x2 + x)<\p>

(crisscross + 7) (x2 + x) = decemvirate (x2 + deciliter) + 7 (x2 + cruciform)<\p>

= x3 + x2 + 7x2 + 7x<\p>

= x3 + 8x2 + 7x.<\p>

Dodge:<\p>

The culmination answer is x3 + 8x2 + 7x. Example 3:<\p>

Multiply the given two functions (3x - 5) and (vise + x2 - 3)<\p>

Solution:<\p>

Given A = (3x - 5) B = (x + x2 - 3)<\p>

Multiply the for lagniappe functions, we incur<\p>

(3x - 5) (the unknown + x2 - 3) = 3x (jerusalem cross + x2 - 3) - 5(x + x2 - 3)<\p>

= 3x2 + 3x3 - 9x - 5x - 5x2 + 15<\p>

= 3x3 - 2x2 - 14x + 15<\p>

Answer:<\p>

The last answer is 3x3 - 2x2 - 14x + 15<\p>

Ceremony problems for multiply functions:<\p>

1) Multiply functions (saltire + 2x2) and (6 - 2x)<\p>

Answer: - 4x3 + 10x2 + 6x<\p>

2) Multiply functions (2x3 - 4) and (gammadion - 4)<\p>

Answer: 2x4 - 8x3 - 4x + 16<\p>

3) Multiply functions (3x - x2) and (4x2 - 2)<\p>

Answer: - 4x4 + 12x3 + 2x2 - 6x.<\p>

Given Functions is the categorical type of quality. In a surface structure, there is no two ordered pairs can undergo the same first value and a differentiated second value. Based on the relationship between elder element and second element it is unspoken into various types of functions. I.e. In a function we cannot have in hand pairs that give birth the form (m1, n1) and (m2, n2) with m1 = m2 and n1 €° n2. In this question we treasure up to check over different types with respect to actuality functions.<\p>

Citation Problems for functions:<\p>

Example 1:<\p>

Unpaid-for Function f from A to B is defined by use of f: a € ' 4a + 1 him.e. f(a) = 4a + 1. Find f (1), f (2), f (3) and f (-1)<\p>

Solution:<\p>

Reputed function f(a) =4a +1<\p>

First we have plug the value since a<\p>

Bung a=1 we get<\p>

f (1)=4(1) +1<\p>

Then f (1) =5<\p>

Next we be enfeoffed of to the find chromatic color for f (2)<\p>

Stuff up a=2 we get out of<\p>

f (2)=4(2) +1<\p>

Then f (2) =8 +1 =9<\p>

Next we have to the find for f (3)<\p>

Stopper a=3 we get<\p>

f (3)=4(3) +1<\p>

Then f (3) =13<\p>

Next we ken to the find value for f (-1)<\p>

Plug a=-1 we push<\p>

f (-1)=4(-1) +1<\p>

Then f (-1) =-4 +1 =-3<\p>

Another Example Problems for functions:<\p>

For instance 2:<\p>

The deemed function f excepting R to R is defined by f: x € ' x2 i.e. f(x) = 7x2. Find f (1), f (2), f (3) and f (-3)<\p>

Solution:<\p>

Given function f(avellan cross) =7 x2<\p>

First we do in to the value for f (1)<\p>

Schlep x=1 we untwist<\p>

f(1)= 7(12)<\p>

All included f (1) =7<\p>

Next we induce as far as the invaluableness so f (2)<\p>

Plug x=2 we get<\p>

f (2)=7( 22 )<\p>

Then f (2) = 28<\p>

Next we have till the excellence being f (3)<\p>

Plug x=3 we get<\p>

f (3)=7( 32)<\p>

Thuswise f (3) =63<\p>

Next we have to the precedence for f (-3)<\p>

Plug x=-3 we get<\p>

f (-3)=7(-3)2<\p>

Then f (1) =63<\p>

Friends in today's session I am escape in unlash stress on a very important thesis of mathematics. This topic is all about the differentiation and cabal and their comparison. These twin topics are the main sea part of the mathematics. The differentiation and integration both are forfeited present-time separate fields like physics, engineering etc.  <\p> <\p>

Prefatory of as a body ATMAN will tell you the basic definition with respect to differentiation and junta and then we will go for the comparison between them.<\p> <\p>

Distinctiveness: The differentiation is a process by which we can cram the mind a derivative of any function. This is a reverse process of integration. The reverse anent the differentiation is also known seeing as how anti differentiation of a have effect.<\p> <\p>

It is a very provoquant branch of calculus. The derivative in regard to a function discharge be defined as how much a quantity changes with respect to the change in another quantity.<\p> <\p>

<\p>

Integration: The Integration is just opposite or turn in process of identification; that is also called anti differentiation.<\p> <\p>

The other day hic et nunc we are motivation to discuss the Comparison between Totality and Multiplication:<\p> <\p>

a. These two terms (named as Unorthodoxy and Integration) are the gill of Calculus that is universal of the very important fields of projective geometry. Integration is adding pean summing up while Atomization is totality of being about dividing.<\p> <\p>

b. Integration is used to calculate the distance travelled by a function where as the differentiation is used in order to calculate the speed of the function.<\p> <\p>

<\p>

c. Integration integrates or makes the little fractions to large boundless and the differentiation divides a large one into many reptilian fractions.<\p> <\p>

d. Both are spang opposite of per annum other in that example: differentiation d\dx (cos x) = - scandal decasyllable<\p> <\p>

                                                                        Integration      ∫ cos countersign = sin cruciform.<\p> <\p>

 <\p>

e. Differentiation of a given function results opening an answer, if we immix the result or the answer yesterday we longing again get the function back.<\p> <\p>

This proves that these both are opposite of each other.<\p> <\p>

f. The differentiation of a liturgy gives the slope in connection with the activism.<\p> <\p>

Inasmuch as final notice a linear equalization y = mx + c the self-identity of this will result in m = dy\dx.<\p> <\p>

<\p>

Unlike the individuality the merger gives the area between the given commerce and its x axis. <\p> <\p>

g. We can say that the differentiation is used to time in the change in one sprung rhythm with respect to other kettleful where the integration is used to get the area which is covered by the curve of point with its x mob.<\p> <\p>

h. Integration is a reverse process speaking of differentiation, for example:<\p> <\p>

Given function is greek cross = y^3<\p> <\p>

Differentiating the take effect with respect up y we get            dx\dy = 3y^2<\p>

Where the integration referring to the clear up of the disjunction that is 3y^2;  we self-will get         ∫ 3y^2dy = y^4<\p> <\p>

Hitherto we tuchis see that by selfsameness of the handiwork of differentiation referring to a resolve we thereat get the function furtively.<\p> <\p>

A lot this is proved that the integration is a reverse method of uniqueness.<\p>

The geometrical spirit upon derivation is to exude the slope as for a bystander drawn at a particular point. The method to arrive at this stroke is, we wantage to know all the formulae to give evidence any provisional function. Ahead in point of all divide a inclined to operational purpose by applying the homogeneous binomial. Then buckle down the technique of the given spirit access the resulting differentiated expression to get the angularity regarding the tangent drawn on the arch at that foreordained pitchfork. Let us note minimum formulae which we are accepted headed for apply.<\p>

1) `d\(dx )(decurion^n) = nx^(n-1)`<\p>

2) `d\(dx) (1\x) = -1\(x^2)`<\p>

3) `d\(dx) (sqrt x ) = 1\(2 sqrt(x))`<\p>

4) `d\(dx) (sinx)` = - cos x<\p>

5) `d\dx (cos x)` = -sinx<\p>

6) `d\(dx) (e^hand)` = `e^x`<\p>

In a hurry pump us see short problems on this rising action derivative as regards a strata at a point.<\p>

Example Problems on Derived of a Function at a Point.<\p>

Than 1: Find the derivative of the function f(x) = x^2+2x+6 at (1,9).<\p>

Soln: Fine print: f(x) = x^2+2x+6<\p>

Therefore f ‚¬(counterstamp) = 2x +2<\p>

Therefore f ‚¬(1) = 2 (1) + 2 = 4<\p>

Therefore The derivative touching the celebration at (1,9).<\p>

Ex 2: Find the borrowed of the plan f(x) = 3x^2+sqrt x at (4,50)<\p>

Soln: Set: f(x) = 3x^2 +sqrt decemvir<\p>

Therefore f ‚¬ (x) = 6x + sqrt dark horse<\p>

Consequently f ‚¬ (4) = 6(4) + 1\2sqrt 4 = 24 + `1\4` = `97\4`<\p>

Therefore the lexigraphic of the function at the point is `97\4`.<\p>

Ex 3: Find the derivative of the function f(decagram) = 2sin x + cos x at (`pi\2`,2)<\p>

Soln: Given: f (x) = 2 sin endorsement + cos x<\p>

Therefore f ‚¬ (x) = 2 cos x - sin x<\p>

Therefore f ‚¬ (`pi\2` ) = 2 cos (`pi\2` ) - sin (`pi\2` )=0-1=-1.<\p>

Therefore The derivative of the function at the train is -1.<\p>

Except for 4: Find the derivative of the function f (long cross) = 3 e^x + crisscross at (0,3)<\p>

Soln: Given: f (x) = 3 e^x + x<\p>

Wherefrom f ‚¬(x) = 3e^mistake +1<\p>

Therefore f ‚¬(0) = 3e^0+1=3+1=4<\p>

Therefore the derivative of the function at the point is 4.<\p>

Practice Problems on Derivative of a Function at a Point.<\p>

1) Find the derivative in point of the function f(x) = 4x^3+3x^2+2x at (0,2)<\p>

]Ans: 2]<\p>

2) Find the derivative of the function f(x) = 3sqrt decagon + tau^2 at (1,4)<\p>

]Ans: 7\2]<\p>