## Friday, November 30, 2018

### ႀကိဳတင္​ပန္​ၾကားျခင္​း

[Unicode]ကြိုတင် ပန်ကြား ခြင်း။
သုံးစွဲသူများ အတွက် data နှင့် အချိန် သက်သာမှုရှိရန် အတွက် text, html, latex code များဖြင့် ရေးရပါသောကြောင့် အချိန် ကြန့်ကြာမှု ရှိခြင်း ကို နားလည် ပေးနိုင်ပါရန် ကြိုတင်၍ အသိပေးအပ ်ပါသည်။

တက္ကသိုလ် ဝင်တန်း သင်္ချာ အတွက် မေးခွန်းဟောင်း များနှင့် အဖြေ ကို ဦးစား ပေး ၍ တင်ပေး ပါမည်။

ကျေးဇူးတင်စွာဖြင့်
Dr Shwe Kyaw (Maths)
Doctor of science, TU Berlin.

[Zawgyi]ႀကိဳတင္ ပန္ၾကား ျခင္း။
သုံးစြဲသူမ်ား အတြက္ data ႏွင့္ အခ်ိန္ သက္သာမႈရွိရန္ အတြက္ text, html, latex code မ်ားျဖင့္ ေရးရပါေသာေၾကာင့္ အခ်ိန္ ၾကန႔္ၾကာမႈ ရွိျခင္း ကို နားလည္ ေပးႏိုင္ပါရန္ ႀကိဳတင္၍ အသိေပးအပ ္ပါသည္။

တကၠသိုလ္ ဝင္တန္း သခ်ၤာ အတြက္ ေမးခြန္းေဟာင္း မ်ားႏွင့္ အေျဖ ကို ဦးစား ေပး ၍ တင္ေပး ပါမည္။

ေက်းဇူးတင္စြာျဖင့္
Dr Shwe Kyaw (Maths)
Doctor of science, TU Berlin.

## Thursday, November 29, 2018

### Sum of k th power of positive integers (Determinant form)

$\newcommand{\D}{\displaystyle} \newcommand{\deta}{\left|\begin{array}{cccccccc}#1\end{array}\right|}$

### 1. (Sum of the first $\D n$ positive integers)

$\sum_{i=1}^{n}i=\frac{1}{2!}\deta{(n+1)^2-(n+1)}$

### 2. (Sum of square of the first $\D n$ positive integers)

$\sum_{i=1}^{n}i^2=\frac{1}{3!}\deta{2&(n+1)^2-(n+1)\\3&(n+1)^3-(n+1)}$

### 3. (Sum of cube of the first $\D n$ positive integers)

$\sum_{i=1}^{n}i^3=\frac{1}{4!}\deta{2&0&(n+1)^2-(n+1)\\3& 3& (n+1)^3-(n+1)\\4&6&(n+1)^4-(n+1)}$

### 4. (Sum of fouth power of the first $\D n$ positive integers)

$\sum_{i=1}^{n}i^4=\frac{1}{5!}\deta{2&0&0&(n+1)^2-(n+1)\\3& 3& 0&(n+1)^3-(n+1)\\4&6& 4&(n+1)^4-(n+1)\\ 5& 10&10& (n+1)^5-(n+1)}$

### 5. (Sum of $\D k^{th}$ power of the first $\D n$ positive integers)

$\sum_{i=1}^{n}i^k=\frac{1}{(k+1)!} \deta{ ^2C_1&0 & 0 &\cdots &\cdots &(n+1)^2-(n+1)\\ ^3C_1&^3C_2 & 0 &\cdots &\cdots &(n+1)^3-(n+1)\\ ^4C_1&^4C_2 & ^4C_3 &\cdots &\cdots &(n+1)^4-(n+1)\\ \vdots& \vdots& \vdots& \vdots& \vdots& \vdots \\ ^{k+1}C_1& ^{k+1}C_2& ^{k+1}C_3& \cdots& ^{k+1}C_{k-1}&(n+1)^{k+1}-(n+1) }$

## Tuesday, November 27, 2018

### Past Questions (AP, GP Series)

$\def\D{\displaystyle} \def\iixi#1#2{\D\left(\begin{array}{c} #1\\#2 \end{array}\right)}$

1.) In an arithmetic sequence, $\D u_1 = 2$ and $\D u_3 = 8.$
(a) Find $\D d.$
(b) Find $\D u_{20}.$
(c) Find $\D S_{20}.$ (Total 6 marks)

2.) In an arithmetic sequence $\D u_1 = 7, u_{20} = 64$ and $\D u_n = 3709.$
(a) Find the value of the common difference.
(b) Find the value of $\D n.$ (Total 5 marks)

3.) Consider the arithmetic sequence 3, 9, 15, ..., 1353.
(a) Write down the common difference.
(b) Find the number of terms in the sequence.
(c) Find the sum of the sequence. (Total 6 marks)

4.) An arithmetic sequence, $\D u_1, u_2, u_3, \ldots ,$ has $\D d = 11$ and $\D u_{27} = 263.$
(a) Find $u_1.$
(b) (i) Given that $\D u_n = 516,$ find the value of $\D n.$
(ii) For this value of $\D n,$ find $\D S_n.$ (Total 6 marks)

5.) The first three terms of an infinite geometric sequence are 32, 16 and 8.
(a) Write down the value of $\D r.$
(b) Find $\D u_6.$
(c) Find the sum to infinity of this sequence. (Total 5 marks)

6.) The $\D n^{th}$ term of an arithmetic sequence is given by $\D u_n = 5 + 2n.$
(a) Write down the common difference.
(b) (i) Given that the $\D n^{th}$ term of this sequence is 115, find the value of $\D n.$
(ii) For this value of $\D n,$ find the sum of the sequence. (Total 6 marks)

7.) In an arithmetic series, the first term is –7 and the sum of the first 20 terms is 620.
(a) Find the common difference.
(b) Find the value of the $\D 78 ^{th}$ term. (Total 5 marks)

8.) In a geometric series, $\D u1 = \frac{1}{81}$ and $\D u_4 = \frac{1}{3} .$
(a) Find the value of $\D r.$
(b) Find the smallest value of $\D n$ for which $\D S_n > 40.$ (Total 7 marks)

9.) (a) Expand $\D \sum_{r=4}^{7} 2^r$ as the sum of four terms.
(b) (i) Find the value of $\D \sum_{r=4}^{30} 2^r.$
(ii) Explain why $\D \sum_{r=4}^{\infty } 2^r$ cannot be evaluated.  (Total 7 marks)

10.) In an arithmetic sequence, $\D S_{40} = 1900$ and $\D u_{40} = 106.$ Find the value of $\D u_1$ and of $\D d.$ (Total 6 marks)

11.) Consider the arithmetic sequence 2, 5, 8, 11, ....
(a) Find $\D u_{101}.$
(b) Find the value of $\D n$ so that $\D u_n = 152.$ (Total 6 marks)

12.) Consider the infinite geometric sequence 3000, - 1800, 1080, -648, … .
(a) Find the common ratio.
(b) Find the 10 th term.
(c) Find the exact sum of the infinite sequence. (Total 6 marks)

13.) Consider the infinite geometric sequence $\D 3, 3(0.9), 3(0.9)^2, 3(0.9)^3, … .$
(a) Write down the 10 th term of the sequence. Do not simplify your answer.
(b) Find the sum of the infinite sequence. (Total 5 marks)

14.) In an arithmetic sequence $\D u_{21} = -37$ and $\D u_4 = -3.$
(a) Find (i) the common difference; (ii) the first term.
(b) Find $\D S_{10}.$ (Total 7 marks)

15.) Let $\D u_n = 3 - 2n.$
(a) Write down the value of $\D u_1, u_2,$ and $\D u_3.$
(b) Find $\D \sum_{n=1}^{20} (3-2n)$ (Total 6 marks)

16.) A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row, and each successive row of seats has two more seats in it than the previous row.
(a) Calculate the number of seats in the 20th row.
(b) Calculate the total number of seats. (Total 6 marks)

17.) A sum of \$5000 is invested at a compound interest rate of 6.3 % per annum. (a) Write down an expression for the value of the investment after n full years. (b) What will be the value of the investment at the end of five years? (c) The value of the investment will exceed \$ 10 000 after n full years.
(i) Write down an inequality to represent this information.
(ii) Calculate the minimum value of n. (Total 6 marks)

18.) Consider the infinite geometric sequence 25, 5, 1, 0.2, … .
(a) Find the common ratio.
(b) Find (i) the 10 th term; (ii) an expression for the n th term.
(c) Find the sum of the infinite sequence. (Total 6 marks)

19.) The first four terms of a sequence are 18, 54, 162, 486.
(a) Use all four terms to show that this is a geometric sequence.
(b) (i) Find an expression for the n th term of this geometric sequence.
(ii) If the n th term of the sequence is 1062 882, find the value of n. (Total 6 marks)

20.) (a) Write down the first three terms of the sequence $\D u_n = 3n,$ for $\D n\ge 1.$
(b) Find (i) $\D \sum_{n=1}^{20} 3n$
(ii) $\D \sum_{n=21}^{100} 3n$  (Total 6 marks)

21.) Consider the infinite geometric series 405 + 270 + 180 +....
(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the fifteenth term of this series.
(c) Find the exact value of the sum of the infinite series. (Total 6 marks)

22.) (a) Consider the geometric sequence -3, 6, -12, 24, ….
(i) Write down the common ratio.
(ii) Find the 15 th term. Consider the sequence $\D x - 3, x +1, 2x + 8,\ldots.$
(b) When $\D x = 5,$ the sequence is geometric.
(i) Write down the first three terms.
(ii) Find the common ratio.
(c) Find the other value of $\D x$ for which the sequence is geometric.
(d) For this value of $\D x,$ find
(i) the common ratio;
(ii) the sum of the infinite sequence. (Total 12 marks)

1.
3,59 610
2. 3,1235
3. 6,226,153228
4. -23,50,12325
5. 1/2,1,64
6. 2,55,3355
7. 4,301
8. 3,8
9.  240,2147483632
10. -11,3
11. 302,51
12. -0.6,-30.2,1875
13. 1.162,30
14. -2,3,-60
15. 1,-1,-3,-360
16. 53,680
17. 5000(1.063^n),6786.35,...,12
18. 0.2,1/78125,...,31.25
19. r=3,18x3^(n-1),11
20. 3,6,9:630,14520
21. 2/3,1.39,1215
22. -2,-49152;2,6,28;3,-5,-16

### Past Questions (Binomial) IB Standard Level

$\newcommand{\D}{\displaystyle} \def\iixi#1#2{\D\left(\begin{array}{c} #1\\#2 \end{array}\right)}$
1.) Consider the expansion of $\D (x + 2)^{11}.$
(a) Write down the number of terms in this expansion.
(b) Find the term containing $\D x^2.$ (Total 5 marks)

2.) (a) Expand $\D (2 + x)^4$ and simplify your result.
(b) Hence, find the term in $\D x^2$ in $\D (2 + x)^4\left(1+\frac{1}{x^2}\right).$ (Total 6 marks)

3.) Find the term in $\D x^4$ in the expansion of $\D \left(3x^2-\frac{2}{x}\right)^5$. (Total 6 marks)

4.) The fifth term in the expansion of the binomial $\D (a + b)^n$ is given by $\D \iixi{10}{4}p^6(2q)^4.$
(a) Write down the value of $\D n$.
(b) Write down $\D a$ and $\D b,$ in terms of $\D p$ and/or $\D q.$
(c) Write down an expression for the sixth term in the expansion.  (Total 6 marks)

5.) Let $\D f(x) = x^3- 4x + 1.$
(a) Expand $\D (x + h)^3.$

6.) Find the term in $\D x^3$ in the expansion of $\D \left(\frac{2}{3}x-3\right)^8.$ (Total 5 marks)

7.) (a) Expand $\D (x-2)^4$ and simplify your result.
(b) Find the term in $\D x^3$ in $\D (3x + 4)(x-2)^4.$  (Total 6 marks)

8.) Consider the expansion of the expression $\D (x^3-3x)^6.$
(a) Write down the number of terms in this expansion.
(b) Find the term in $\D x^{12}.$ (Total 6 marks)

9.) One of the terms of the expansion of $\D (x + 2y)^{10}$ is $\D ax^8 y^2.$ Find the value of a. (Total 6 marks)

10.) (a) Expand $\D \left(e+\frac{1}{e}\right)^4$ in terms of $\D e.$
(b) Express $\D \left(e+\frac{1}{e}\right)^4+\left(e-\frac{1}{e}\right)^4$ as the sum of three terms.  (Total 6 marks)

11.) Consider the expansion of $\D (x^2- 2)^5.$
(a) Write down the number of terms in this expansion.
(b) The first four terms of the expansion in descending powers of $\D x$ are $x^{10} -10x^8 + 40x^6 + Ax^4 + ...$ Find the value of A. (Total 6 marks)

12.) Given that $\D \left(3+\sqrt{7}\right)^3=p+\sqrt{q}$ where $\D p$ and $\D q$ are integers, find (a) $\D p;$
(b) $\D q.$ (Total 6 marks)

13.) When the expression $\D (2 + ax)^{10}$ is expanded, the coefficient of the term in $\D x^3$ is 414720. Find the value of $\D a.$ (Total 6 marks)

14.) Find the term containing $\D x^3$ in the expansion of $\D (2- 3x)^8.$ (Total 6 marks)

15.) Find the term containing $\D x^{10}$ in the expansion of $\D (5 + 2x^2)^7.$ (Total 6 marks)

16.) Complete the following expansion. $\D (2 + ax)^4 = 16 + 32ax +\cdots$ (Total 6 marks)

17.) Consider the expansion of $\D \left(3x^2-\frac{1}{x}\right)^9.$
(a) How many terms are there in this expansion?
(b) Find the constant term in this expansion. (Total 6 marks)

18.) Find the coefficient of $\D x^3$ in the expansion of $\D (2- x)^5.$ (Total 6 marks)

19.) Use the binomial theorem to complete this expansion. $\D (3x + 2y)^4 = 81x^4 + 216x^3 y + \cdots$ (Total 4 marks)

20.) Consider the binomial expansion . $(1+x)^4=1+\iixi{4}{1}x+ \iixi{4}{2}x^2 +\iixi{4}{3}x^3+x^4.$
(a) By substituting $\D x = 1$ into both sides, or otherwise, evaluate $\D \iixi{4}{1}+ \iixi{4}{2} +\iixi{4}{3}$
(b) Evaluate $\D \iixi{9}{1}+ \iixi{9}{2}+ \iixi{9}{3}+ \iixi{9}{4}+ \iixi{9}{5}+ \iixi{9}{6}+ \iixi{9}{7}+ \iixi{9}{8}.$ (Total 4 marks)

21.) Determine the constant term in the expansion of $\D \left(x-\frac{2}{x^2}\right)^9.$ (Total 4 marks)

22.) Find the coefficient of $\D a^5b^7$ in the expansion of $\D (a + b)^{12}.$ (Total 4 marks)

23.) Find the coefficient of $\D x^5$ in the expansion of $\D (3x - 2)^8.$ (Total 4 marks) 24.) Find the coefficient of $\D a^3b^4$ in the expansion of $(5a + b)^7.$ (Total 4 marks)

1. 12,28160x^2
2. 16 32 24 8 1
25x^2
3. 1080x^4
4. n=10,a=p,b=2q,10C5p^5(2q)^5
5. x^3+3x^2h+3xh^2+h^3
6. -4032x^3
7. x^4-8x^3+24x^2-32x+16;40x^3
8. 7;-540x^{12}
9. 180
10. e^4+4e^2+6+4/e^2+1/e^4
2e^4+12+2?e^4
11. 6;A=-80
12. p=90,q=34
13. 3
14. -48384x^3
15. 16800x^{10}
16. ...+24a^2x^2+8a^3x^3+a^4x^4
17. 10;2268
18. -40
19. ...+216x^2y^2+96xy^3+16y^4
20. 14;510
21. -672
22. 792
23. -108864
24. 4375

## Method of difference

### Example 1

$\newcommand{\D}{\displaystyle}$ Find $\D \sum_{k=1}^{n}\frac{1}{k(k+1)}.$
Let $\D u_k=\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}.$
\begin{eqnarray} u_1 &=&\frac{1}{1}-\frac{1}{2}\\ u_2 &=&\frac{1}{2}-\frac{1}{3}\\ u_3 &=&\frac{1}{3}-\frac{1}{4}\\ \vdots &=& \vdots\\ u_n &=&\frac{1}{n}-\frac{1}{n+1} \end{eqnarray} By adding, $u_1+u_2+\cdots+u_n=1-\frac{1}{n+1}$ $\newcommand{\iixi}{\left(\begin{array}{c}#1\\#2\end{array}\right)}$

### Proposition:

For any positive integer $n$ and $k$ $(n+1)^k-1=\iixi{k}{1}\sum_{i=1}^{n}i^{k-1} +\iixi{k}{2}\sum_{i=1}^{n}i^{k-2}+\cdots+\iixi{k}{k}n .$

### Proof:

\begin{eqnarray*} (n+1)^k&=&n^k+\iixi{k}{1} n^{k-1}+\iixi{k}{2}n^{k-2}+\cdots+\iixi{k}{k}\\ (n+1)^k-n^k&=&\iixi{k}{1} n^{k-1}+\iixi{k}{2}n^{k-2}+\cdots+\iixi{k}{k}\\ \end{eqnarray*} \begin{eqnarray*} n=1:&2^k-1^k=&\iixi{k}{1} 1^{k-1}+\iixi{k}{2}1^{k-2}+\cdots+\iixi{k}{k}\\ n=2:&3^k-2^k=&\iixi{k}{1} 2^{k-1}+\iixi{k}{2}2^{k-2}+\cdots+\iixi{k}{k}\\ n=3:&4^k-3^k=&\iixi{k}{1} 3^{k-1}+\iixi{k}{2}3^{k-2}+\cdots+\iixi{k}{k}\\ \vdots&\vdots&\vdots\\ &(n+1)^k-n^k=&\iixi{k}{1} n^{k-1}+\iixi{k}{2}n^{k-2}+\cdots+\iixi{k}{k}\\ \end{eqnarray*} By adding $(n+1)^k-1=\iixi{k}{1}\sum_{i=1}^{n}i^{k-1} +\iixi{k}{2}\sum_{i=1}^{n}i^{k-2}+\cdots+\iixi{k}{k}n$

### Example 2 (Sum of the first $n$ natural numbers)

$S_n=\sum_{i=1}^{n}i=\frac{n(n+1)}{2}.$ By considering $k=2$, in above proposition, we get $(n+1)^2-1=\iixi{2}{1}\sum_{i=1}^{n}i+\iixi{2}{2}n$ Thus, \begin{eqnarray*} n^2+2n+1-1&=&2S_n+n\\ 2S_n&=&n^2+n=n(n+1)\\ S_n&=&\frac{n(n+1)}{2} \end{eqnarray*}

### Example 3 (Sum of the square of the first $n$ natural numbers, denoted by $S_n^{(2)}$)

$S_n^{(2)}=\sum_{i=1}^{n}i^2=\frac{n(n+1)(2n+1)}{6}.$ By considering $k=3$, in above proposition, we get $(n+1)^3-1=\iixi{3}{1}\sum_{i=1}^{n}i^2+ \iixi{3}{2}\sum_{i=1}^{n}i+\iixi{3}{3}n$ Thus, \begin{eqnarray*} n^3+3n^2+3n+1-1&=&3S_n^{(2)}+3S_n+n\\ n^3+3n^2+3n&=&3S_n^{(2)}+\frac{3n(n+1)}{2}+n\\ 3S_n^{(2)}&=&n^3+3n^2+3n-n-\frac{3n(n+1)}{2}\\ S_n^{(2)}&=&\frac{n(n+1)(2n+1)}{6} \end{eqnarray*}

### Example 4 (Sum of the square of the first $n$ even numbers)

$2^2+4^2+\cdots+(2n)^2=\frac{2n(n+1)(2n+1)}{3}.$ By considering $k=3$, in above proposition, we get $(n+1)^3-1=\iixi{3}{1}\sum_{i=1}^{n}i^2+ \iixi{3}{2}\sum_{i=1}^{n}i+\iixi{3}{3}n$ Let $u_i=2i,i=1,2,\ldots,n$. Then $(u_i)^2=4i^2.$ Thus, $\D i^2=\frac{(u_i)^2}{4}.$ \begin{eqnarray*} n^3+3n^2+3n+1-1&=&3\sum_{i=1}^{n}\frac{(u_i)^2}{4}+3S_n+n\\ n^3+3n^2+3n&=&\frac{3}{4}\sum_{i=1}^{n}(u_i)^2+\frac{3n(n+1)}{2}+n\\ \frac{3}{4}\sum_{i=1}^{n}(u_i)^2&=&n^3+3n^2+3n-n-\frac{3n(n+1)}{2}\\ \sum_{i=1}^{n}(2i)^2&=&\frac{2n(n+1)(2n+1)}{3} \end{eqnarray*}

### Example 5 (Sum of the square of the first $n$ odd numbers)

$1^2+3^2+\cdots+(2n-1)^2=\frac{n(2n+1)(2n-1)}{3}$ By considering $k=3$, in above proposition, we get $(n+1)^3-1=\iixi{3}{1}\sum_{i=1}^{n}i^2+ \iixi{3}{2}\sum_{i=1}^{n}i+\iixi{3}{3}n$ Let $u_i=2i-1,i=1,2,\ldots,n.$ Then $(u_i)^2=4i^2-4i+1.$ Thus, $\D i^2=\frac{(u_i)^2+4i-1}{4}.$ \begin{eqnarray*} n^3+3n^2+3n+1-1&=&3\sum_{i=1}^{n}\frac{(u_i)^2+4i-1}{4}+3S_n+n\\ n^3+3n^2+3n&=&\frac{3}{4}\sum_{i=1}^{n}(u_i)^2+\frac{3}{4}\sum_{i=1}^{n}(4i-1)+\frac{3n(n+1)}{2}+n\\ \frac{3}{4}\sum_{i=1}^{n}(u_i)^2&=&n^3+3n^2+3n-n-\frac{3n(n+1)}{2}-\frac{3}{4}\sum_{i=1}^{n}(4i-1)\\ \sum_{i=1}^{n}(2i-1)^2&=&\frac{n(2n+1)(2n-1)}{3} \end{eqnarray*}

### Example 6

Show that $1\times2+2\times3+3\times4\cdots +n(n+1)=\frac{n(n+1)(n+2)}{3}$ Proof: Let $\D u_i=i(i+1)$. By considering $k=3$, in above proposition, we get $(n+1)^3-1=\iixi{3}{1}\sum_{i=1}^{n}i^2+ \iixi{3}{2}\sum_{i=1}^{n}i+\iixi{3}{3}n$ \begin{eqnarray*} n^3+3n^2+3n+1-1&=&3\sum_{i=1}^{n}(i^2+i)+n\\ n^3+3n^2+3n&=&3\sum_{i=1}^{n}u_i+n\\ 3\sum_{i=1}^{n}u_i&=&n^3+3n^2+3n-n\\ \sum_{i=1}^{n}u_i&=&\frac{1}{3}(n^3+3n^2+2n)\\ &=&\frac{n(n+1)(n+2)}{3} \end{eqnarray*}

### Exam Practice

By using the formula $1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6},$
(1)show: $1^2+3^2+5^2+\cdots+(2n-1)^2=\frac{n(2n+1)(2n-1)}{3}.$
(2)verify: $2^2+4^2+6^2+\cdots+(2n)^2=\frac{2n(n+1)(2n+1)}{3}.$
(3)prove: $1\times2+2\times3+3\times4\cdots +n(n+1)=\frac{n(n+1)(n+2)}{3}$
(4) Evaluate: (Product of the same term of an AP) $(-5)^2+(-2)^2+1^2+4^2+\cdots+ \mbox{10 terms}$ (Hint: Let $\D u_i=-8+3i$. Then $\D u_i^2=\cdots$ )
(5) Find: (Product of the consecutive terms of an AP) $(-2)(1)+(1)(4)+(4)(7)+\cdots \mbox{10 terms}$ (Hint: Let $\D u_i=(-5+3i)(-2+3i)$)
(6) Calculate: (Product of corresponding  terms of two APs)
$(-3)(5)+(-1)(8)+(1)(11)+\cdots +\mbox{10 terms}$

## H-factor Method for differentation by using first principle

### Example 1 (h-factor method)

Find $\displaystyle f'(x)$, if $f(x)=\sqrt{x}.$ \begin{eqnarray*} f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\ &=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(x+h)-x}\\ &=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h})^2-(\sqrt{x})^2}\\ &=&\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}\\ &=& \lim_{h\to 0} \frac{1}{(\sqrt{x+h}+\sqrt{x})}\\ &=& \frac{1}{\sqrt{x}+\sqrt{x}}\\ &=&\frac{1}{2\sqrt{x}} \end{eqnarray*}

### Example 2 (h-factor method)

Find $\displaystyle f'(x)$, if $f(x)=\sqrt{x}.$ \begin{eqnarray*} f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\ &=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(x+h)-x}\\ &=&\lim_{h\to 0}\frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h})^3-(\sqrt{x})^3}\\ &=&\lim_{h\to 0} \frac{\sqrt{x+h}-\sqrt{x}}{(\sqrt{x+h}-\sqrt{x})((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \lim_{h\to 0} \frac{1}{((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \frac{1}{\sqrt{x^2}+\sqrt{x^2} +\sqrt{x^2}}\\ &=&\frac{1}{3\sqrt{x^2}} \end{eqnarray*}

### Example 3 (h-factor method)

Find $\displaystyle f'(x)$, if $f(x)=\sqrt{x^2}.$ \begin{eqnarray*} f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\ &=&\lim_{h\to 0}\frac{(\sqrt{x+h})^2-(\sqrt{x})^2}{(x+h)-x}\\ &=&\lim_{h\to 0}\frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{(\sqrt{x+h})^3-(\sqrt{x})^3}\\ &=&\lim_{h\to 0} \frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})}{(\sqrt{x+h}-\sqrt{x})((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \lim_{h\to 0} \frac{\sqrt{x+h}+\sqrt{x}}{((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \frac{\sqrt{x}+\sqrt{x}}{\sqrt{x^2}+\sqrt{x^2} +\sqrt{x^2}}\\ &=&\frac{2\sqrt{x}}{3\sqrt{x^2}}=\frac{2}{3\sqrt{x}} \end{eqnarray*}

### Example 4 (h-factor method)

Find $\displaystyle f'(x)$, if $f(x)=\sqrt{x^4}.$ \begin{eqnarray*} f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\\ &=&\lim_{h\to 0}\frac{(\sqrt{x+h})^4-(\sqrt{x})^4}{(x+h)-x}\\ &=&\lim_{h\to 0}\frac{((\sqrt{x+h})^2-(\sqrt{x})^2)((\sqrt{x+h})^2+(\sqrt{x})^2)}{(\sqrt{x+h})^3-(\sqrt{x})^3}\\ &=&\lim_{h\to 0} \frac{(\sqrt{x+h}-\sqrt{x})(\sqrt{x+h}+\sqrt{x})((\sqrt{x+h})^2+(\sqrt{x})^2)}{(\sqrt{x+h}-\sqrt{x})((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \lim_{h\to 0} \frac{(\sqrt{x+h}+\sqrt{x})((\sqrt{x+h})^2+(\sqrt{x})^2)}{((\sqrt{x+h})^2+ \sqrt{x+h}\sqrt{x} +(\sqrt{x})^2)}\\ &=& \frac{(\sqrt{x}+\sqrt{x})((\sqrt{x})^2+(\sqrt{x})^2)}{\sqrt{x^2}+\sqrt{x^2} +\sqrt{x^2}}\\ &=&\frac{2\sqrt{x}\times2\sqrt{x^2}}{3\sqrt{x^2}}=\frac{4\sqrt{x}}{3} \end{eqnarray*}