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RD Sharma Solution for Class 12 Science Mathematics Chapter 12 - Higher Order Derivatives

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 12 - Higher Order Derivatives.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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Exercise/Page

RD Sharma Solution for Class 12 Science Mathematics Chapter 12 - Higher Order Derivatives Page/Excercise 12.1

Solution 1(i)

Solution 1(ii)

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

Solution 1(iv)


Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 1(viii)

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

Solution 1(ix)

Solution 2

Solution 3

Solution 4

Solution 5

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

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10


Solution 11


Solution 12

Solution 13

Solution 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

Solution 15

Solution 16

Solution 17


Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

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.

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


Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49


Solution 50

Solution 51

Solution 52

RD Sharma Solution for Class 12 Science Mathematics Chapter 12 - Higher Order Derivatives Page/Excercise MCQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Correct option: (d)

Solution 6

Solution 7

  

Solution 8

Solution 9

Solution 10

Solution 11

 

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

 Question is incorrect.

   

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

RD Sharma Solution for Class 12 Science Mathematics Chapter 12 - Higher Order Derivatives Page/Excercise 12VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

y equals x plus e to the power of x
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 y over denominator d x end fraction equals 1 plus e to the power of x
T h u s comma space fraction numerator d x over denominator d y end fraction equals fraction numerator 1 over denominator 1 plus e to the power of x end fraction
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 e q u a t i o n space w i t h space r e s p e c t space t o space y comma space w e space h a v e comma
fraction numerator d squared x over denominator d y squared end fraction equals fraction numerator minus e to the power of x over denominator open parentheses 1 plus e to the power of x close parentheses squared end fraction cross times fraction numerator d x over denominator d y end fraction
equals fraction numerator minus e to the power of x over denominator open parentheses 1 plus e to the power of x close parentheses cubed end fraction

Solution 8

Solution 9

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 12 - Higher Order Derivatives for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 12 - Higher Order Derivatives.

Text Book Solutions

CBSE XII Science - Mathematics

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