Chapter 12 Higher Order Derivatives Ex 12.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Find the second order derivative of log(sinx).Solution 1(iii)

L e t space y equals log left parenthesis sin x right parenthesis D i f f e r e n t i a t i n g space w i t h space r e p e c t space t o space x comma space w e space g e t comma fraction numerator d y over denominator d x end fraction equals fraction numerator cos x over denominator sin x end fraction A g a i n space d i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space x comma space w e space g e t comma fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus sin x cross times sin x minus cos x cross times cos x over denominator sin squared x end fraction rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus sin squared x minus cos squared x over denominator sin squared x end fraction rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus open parentheses sin squared x plus cos squared x close parentheses over denominator sin squared x end fraction rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus 1 over denominator sin squared x end fraction rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals minus cos e c squared x

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 1(ix)

Solution 1(ix)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

If x equals a open parentheses theta minus sin theta close parentheses comma space y equals a open parentheses 1 plus cos theta close parentheses, find fraction numerator d squared y over denominator d x squared end fractionSolution 14

x equals a open parentheses theta minus sin theta close parentheses ; space y equals a open parentheses 1 plus cos theta close parentheses D i i f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n s space w i t h space r e s p e c t space t o space theta comma space w e space g e t comma fraction numerator d x over denominator d theta end fraction equals a open parentheses 1 minus cos theta close parentheses space space space... left parenthesis 1 right parenthesis fraction numerator d y over denominator d theta end fraction equals a open parentheses minus sin theta close parentheses space space space space space space space... left parenthesis 2 right parenthesis D i v i d i n g space e q u a t i o n space left parenthesis 2 right parenthesis space b y space left parenthesis 1 right parenthesis comma space w e space h a v e comma fraction numerator d y over denominator d x end fraction equals fraction numerator a open parentheses minus sin theta close parentheses over denominator a open parentheses 1 minus cos theta close parentheses end fraction space equals fraction numerator minus sin theta over denominator 1 minus cos theta end fraction D i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space theta comma space w e space h a v e comma fraction numerator d open parentheses fraction numerator d y over denominator d x end fraction close parentheses over denominator d theta end fraction equals fraction numerator open parentheses 1 minus cos theta close parentheses open parentheses minus cos theta close parentheses plus sin theta open parentheses sin theta close parentheses over denominator open parentheses 1 minus cos theta close parentheses squared end fraction equals fraction numerator minus cos theta plus cos squared theta plus sin squared theta over denominator open parentheses 1 minus cos theta close parentheses squared end fraction equals fraction numerator 1 minus cos theta over denominator open parentheses 1 minus cos theta close parentheses squared end fraction fraction numerator d open parentheses fraction numerator d y over denominator d x end fraction close parentheses over denominator d theta end fraction equals fraction numerator 1 over denominator 1 minus cos theta end fraction... left parenthesis 3 right parenthesis D i v i d i n g space e q u a t i o n space left parenthesis 3 right parenthesis space b y space left parenthesis 1 right parenthesis comma space w e space h a v e comma fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator 1 over denominator 1 minus cos theta end fraction cross times fraction numerator 1 over denominator a open parentheses 1 minus cos theta close parentheses end fraction equals fraction numerator 1 over denominator a open parentheses 1 minus cos theta close parentheses squared end fraction equals fraction numerator 1 over denominator a open parentheses 2 sin squared begin display style theta over 2 end style close parentheses squared end fraction equals fraction numerator 1 over denominator 4 a sin to the power of 4 open parentheses theta over 2 close parentheses end fraction equals fraction numerator 1 over denominator 4 a end fraction cos e c to the power of 4 open parentheses theta over 2 close parentheses

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

begin mathsize 12px style If space straight y equals straight e to the power of acos to the power of negative 1 end exponent straight x end exponent comma space show space that space open parentheses 1 minus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx minus straight a squared straight y equals 0 end style

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

I f space y equals cos e c to the power of minus 1 end exponent x comma space x greater than 1 comma space t h e n space s h o w space t h a t space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction plus open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals 0

Solution 42

W e space k n o w space t h a t comma space fraction numerator d over denominator d x end fraction open parentheses cos e c to the power of minus 1 end exponent x close parentheses equals fraction numerator minus 1 over denominator open vertical bar x close vertical bar square root of x squared minus 1 end root end fraction L e t space y equals cos e c to the power of minus 1 end exponent x fraction numerator d y over denominator d x end fraction equals fraction numerator minus 1 over denominator open vertical bar x close vertical bar square root of x squared minus 1 end root end fraction S i n c e space x greater than 1 comma space open vertical bar x close vertical bar equals x T h u s comma fraction numerator d y over denominator d x end fraction equals fraction numerator minus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 1 right parenthesis D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator x begin display style fraction numerator 2 x over denominator 2 square root of x squared minus 1 end root end fraction end style plus square root of x squared minus 1 end root over denominator x squared open parentheses x squared minus 1 close parentheses end fraction equals fraction numerator begin display style fraction numerator x squared over denominator square root of x squared minus 1 end root end fraction end style plus square root of x squared minus 1 end root over denominator x squared open parentheses x squared minus 1 close parentheses end fraction equals fraction numerator x squared plus x squared minus 1 over denominator x squared open parentheses x squared minus 1 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction equals fraction numerator 2 x squared minus 1 over denominator x squared open parentheses x squared minus 1 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction T h u s comma space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator 2 x squared minus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 2 right parenthesis S i m i l a r l y comma space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator minus 2 x squared plus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 3 right parenthesis T h u s comma space f r o m space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction plus open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x squared minus 1 over denominator x square root of x squared minus 1 end root end fraction plus open parentheses fraction numerator minus 2 x squared plus 1 over denominator x square root of x squared minus 1 end root end fraction close parentheses equals 0 H e n c e space p r o v e d.

Question 43

I f space x equals cos t plus log tan t over 2 comma space y equals sin t comma space t h e n space f i n d space t h e space v a l u e space o f space fraction numerator d squared y over denominator d t squared end fraction space a n d space fraction numerator d squared y over denominator d x squared end fraction space a t space t equals straight pi over 4.

Solution 43

G i v e n space t h a t comma space x equals cos t plus log tan t over 2 comma space y equals sin t D i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space t comma space w e space h a v e comma fraction numerator d x over denominator d t end fraction equals minus sin t plus fraction numerator space 1 over denominator tan begin display style t over 2 end style end fraction cross times s e c squared t over 2 cross times 1 half equals minus sin t plus fraction numerator space 1 over denominator begin display style fraction numerator sin begin display style t over 2 end style over denominator cos t over 2 end fraction end style end fraction cross times fraction numerator 1 over denominator cos squared begin display style t over 2 end style end fraction cross times 1 half equals minus sin t plus fraction numerator space 1 over denominator begin display style fraction numerator sin begin display style t over 2 end style over denominator cos t over 2 end fraction end style end fraction cross times fraction numerator 1 over denominator cos squared begin display style t over 2 end style end fraction cross times 1 half equals minus sin t plus fraction numerator space 1 over denominator 2 sin t over 2 cos t over 2 end fraction equals minus sin t plus fraction numerator space 1 over denominator sin t end fraction equals fraction numerator 1 minus sin squared t over denominator sin t end fraction equals fraction numerator cos squared t over denominator sin t end fraction equals cos t cross times c o t t
N o w space f i n d space t h e space v a l u e space o f space fraction numerator d y over denominator d t end fraction : fraction numerator d y over denominator d t end fraction equals cos t T h u s comma space fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d t end fraction cross times fraction numerator d t over denominator d x end fraction equals cos t cross times fraction numerator 1 over denominator cos t cross times c o t t end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals tan t S i n c e space fraction numerator d y over denominator d t end fraction equals cos t comma space w e space h a v e space fraction numerator d squared y over denominator d t squared end fraction equals minus sin t A t space t equals straight pi over 4 comma space open parentheses fraction numerator d squared y over denominator d t squared end fraction close parentheses subscript t equals straight pi over 4 end subscript equals minus sin open parentheses straight pi over 4 close parentheses equals fraction numerator minus 1 over denominator square root of 2 end fraction
fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator begin display style fraction numerator d over denominator d t end fraction end style open parentheses begin display style fraction numerator d y over denominator d x end fraction end style close parentheses over denominator begin display style fraction numerator d x over denominator d t end fraction end style end fraction equals fraction numerator begin display style fraction numerator d over denominator d t end fraction end style open parentheses begin display style tan t end style close parentheses over denominator begin display style cos t cross times c o t t end style end fraction equals fraction numerator s e c squared t over denominator begin display style cos t cross times c o t t end style end fraction equals fraction numerator s e c squared t over denominator begin display style cos t cross times fraction numerator begin display style cos t end style over denominator sin t end fraction end style end fraction equals fraction numerator s e c squared t over denominator begin display style cos squared t end style end fraction cross times sin t equals s e c to the power of 4 t cross times sin t T h u s comma space open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses subscript t equals straight pi over 4 end subscript equals s e c to the power of 4 open parentheses straight pi over 4 close parentheses cross times sin straight pi over 4 equals 2

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 9

If   prove that   and   Solution 9

Given: 

Differentiating ‘x’ w.r.t   we get

Differentiating ‘y’ w.r.t   we get

Dividing (ii) by (i), we get

 … (iii)

Differentiating above equation w.r.t x, we get

Hence,   Question 48

If   find   Solution 48

Given: 

Differentiating ‘x’ w.r.t t, we get

Differentiating ‘y’ w.r.t t, we get

Dividing (ii) by (i), we get

Differentiating above equation w.r.t x, we get

Hence, 

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