Find the best tutors and institutes for Class 12 Tuition
Q1:
Find $\frac{dy}{dx}$ in the following: $2x + 3y = \sin x$
Solution :
Step 1: Identifying the Mathematical Structure
The given equation is an implicit function of the form $f(x, y) = c$, where $2x + 3y = \sin x$. To find the derivative $\frac{dy}{dx}$, we must differentiate both sides of the equation with respect to $x$ [Per the Sum Rule and Chain Rule of Calculus].
Step 2: Differentiating Both Sides with Respect to $x$
We apply the operator $\frac{d}{dx}$ to each term independently:
$$\frac{d}{dx}(2x) + \frac{d}{dx}(3y) = \frac{d}{dx}(\sin x)$$
Step 3: Applying Derivative Rules
- For the first term, apply the Power Rule: $\frac{d}{dx}(ax^n) = anx^{n-1}$. Thus, $\frac{d}{dx}(2x) = 2(1) = 2$.
- For the second term, apply the constant multiple rule and the implicit differentiation rule: $\frac{d}{dx}(3y) = 3 \cdot \frac{dy}{dx}$.
- For the right-hand side, apply the standard trigonometric derivative: $\frac{d}{dx}(\sin x) = \cos x$.
Step 4: Algebraic Manipulation to Isolate $\frac{dy}{dx}$
Substituting the results from Step 3 into our equation:
$$2 + 3\frac{dy}{dx} = \cos x$$
Subtracting $2$ from both sides [Per the Subtraction Property of Equality]:
$$3\frac{dy}{dx} = \cos x - 2$$
Dividing by $3$ to isolate the derivative:
$$\frac{dy}{dx} = \frac{\cos x - 2}{3}$$
Visualizing the Gradient Function
Final Solution: $\frac{dy}{dx} = \frac{\cos x - 2}{3}$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.3
- Q2: Find $\frac{dy}{dx}$ in the following: $2x + 3y = \sin y$
- Q3: Find $\frac{dy}{dx}$ in the following: $ax + by^2 = \cos y$
- Q4: Find $\frac{dy}{dx}$ in the following: $xy + y^2 = \tan x + y$
- Q5: Find $\frac{dy}{dx}$ in the following: $x^2 + xy + y^2 = 100$
- Q6: Find $\frac{dy}{dx}$ in the following: $x^3 + x^2y + xy^2 + y^3 = 81$
- Q7: Find $\frac{dy}{dx}$ in the following: $\sin^2 y + \cos xy = \kappa$
- Q8: Find $\frac{dy}{dx}$ in the following: $\sin^2 x + \cos^2 y = 1$
- Q9: Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1} \left(\frac{2x}{1+x^2}\right)$
- Q10: Find $\frac{dy}{dx}$ in the following: $y = \tan^{-1}\left(\frac{3x - x^3}{1 - 3x^2}\right), -\frac{1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}$
- Q11: Find $\frac{dy}{dx}$ in the following: $y = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$
- Q12: Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1}\left(\frac{1-x^2}{1+x^2}\right), 0 < x < 1$
- Q13: Find $\frac{dy}{dx}$ in the following: $y = \cos^{-1}\left(\frac{2x}{1+x^2}\right), -1 < x < 1$
- Q14: Find $\frac{dy}{dx}$ in the following: $y = \sin^{-1}(2x\sqrt{1-x^2}), -\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}$
- Q15: Find $\frac{dy}{dx}$ in the following: $y = \sec^{-1}\left(\frac{1}{2x^2-1}\right), 0 < x < \frac{1}{\sqrt{2}}$
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Continuity and Differentiability
Top Tutor
Darbhanga City, Darbhanga
1 yrs of
Exp
3 students
45 hrs of classes completed
600
per hour
Online Classes
I am a TESOL and TEFL certified English tutor with a Master’s degree in English Literature dedicated to helping learners develop confidence and communicate effectively in English. My teaching approach is patient, structured and interactive with a focus on practical communication, regular practice and steady progress. I work with adults and young learners who want to improve their Spoken English, fluency, vocabulary, grammar, and overall communication skills. My lessons are personalised to every learner's level, goals and areas of difficulty, creating a supportive space where they can practice, speaking confidently and naturally.
Studying Class 12 Core English with her was an amazing experience. She completely knows her subject and brings immense clarity to every single lesson. Her explanations made even the most difficult chapters incredibly easy to understand. I highly recommend her classes to any student looking to ace their boards!
Top Tutor
Laxmi Vihar, Hisar
4 yrs of
Exp
11 students
267 hrs of classes completed
800
per hour
Online Classes
I'm a mathematics' teacher. My goal is to teach you how you can think out a solution rather than roboting your way out. We will dive deep into concepts of mathematics and come out as a well learned and scrupulous student. I am ready to clear your hurdles with most accurate way. I have taught at many academies for JEE preparations. In addition to it I have taught online in UrbanPro from past 4 years.
Suman is an awesome teacher and connects well with the student and gets them to grasp concepts and be interested in the subject. Highly recommend him for math.
Top Tutor
Vishnu Puri Colony, Indore
8 yrs of
Exp
2 students
600
per hour
Online Classes
I have been guiding students past 8 years from now and it let helps my students to score above 90% and perform well in class 11 ,12 ,jee mains and neet..
She is amazing and very hardworking. she understands all my problems in fact clear my all doubts. She encourage us and give us strength for top
Top Tutor
Sector-18, Faridabad
1 yrs of
Exp
7 students
1108 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I have approximately 3 years of experience of teaching Mathematics across different boards including CBSE, IGCSE and state board in reputed schools like DPS International. I am post graduate in statistics and honors degree in mathematics from reputed universities like DU. I have helped many students to overcome their fear of Mathematics and Statistics. My aim is to develop a genuine love for the subject in students.
Maam is good and was able to resolve my queries. She is very calm explains the chapter thoroughly. It was just one class with her but we are going to have more in up coming days.
Super Tutor
Vasantha Vallabha Nagar, Bangalore
20 yrs of
Exp
38 students
2481 hrs of classes completed
400
per hour
Online Classes
I am a teacher with over 25 years of experience. Teaching is my passion. I teach across ICSE, ISC, IGCSE , AS&A Levels. I keep producing excellent results year after year through my vast knowledge&experience in the subjects I teach.
Nice teacher. Very punctual , always follow her schedule. Hold on her subject ie economics. Highly recommended.
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Continuity and Differentiability EXERCISE 5.3 worksheets
Download Now
