Class 12 RS Aggarwal Chapter-10 Differentiation Solutions - Free PDF Download
FAQs on RS Aggarwal Class 12 Solutions Chapter-10 Differentiation
1. What are the key topics and exercises covered in RS Aggarwal Class 12 Maths Chapter 10 on Differentiation?
RS Aggarwal Class 12 Chapter 10 provides comprehensive practice in Differentiation. The chapter is structured with multiple exercises that cover the following key topics:
- Differentiation of inverse trigonometric functions.
- Derivatives of exponential and logarithmic functions.
- The application of the Chain Rule for functions of a function.
- Differentiation of implicit functions.
- The method of logarithmic differentiation for complex expressions.
- Derivatives of functions in parametric forms.
- Calculation of second-order derivatives.
Each exercise focuses on a specific method or type of function, allowing students to build mastery step-by-step.
2. How does RS Aggarwal's Chapter 10 align with the NCERT curriculum and the CBSE Class 12 syllabus for the 2025-26 session?
Chapter 10 of RS Aggarwal directly aligns with Unit III (Calculus) of the CBSE Class 12 Maths syllabus for 2025-26, specifically corresponding to the NCERT chapter on 'Continuity and Differentiability'. While NCERT establishes the core concepts, RS Aggarwal offers a more extensive set of problems with varying difficulty levels, providing the rigorous practice needed for both board exams and competitive entrance tests.
3. What are the fundamental differentiation formulas I must know for solving problems in RS Aggarwal Chapter 10?
To effectively solve the problems in this chapter, you must be proficient with the following fundamental formulas:
- Algebraic: d/dx (xⁿ) = nxⁿ⁻¹; d/dx (constant) = 0
- Trigonometric: d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec²x
- Inverse Trigonometric: d/dx (sin⁻¹x) = 1/√(1-x²); d/dx (tan⁻¹x) = 1/(1+x²)
- Exponential & Logarithmic: d/dx (eˣ) = eˣ; d/dx (log x) = 1/x
Mastering these, along with the product, quotient, and chain rules, is crucial for success.
4. When solving problems from RS Aggarwal, how do I decide whether to use the Product Rule, Quotient Rule, or Chain Rule?
The choice of rule depends entirely on the structure of the function you need to differentiate:
- Use the Product Rule when the function is a product of two other functions, like f(x) = u(x) · v(x).
- Use the Quotient Rule when the function is a ratio of two functions, such as f(x) = u(x) / v(x).
- Use the Chain Rule for a composite function, where one function is nested inside another, like f(x) = g(h(x)). For example, in sin(x²), the chain rule is necessary because x² is a function inside the sine function.
Often, you may need to use a combination of these rules for more complex expressions.
5. What is a common mistake students make when differentiating implicit functions in RS Aggarwal, and how can it be avoided?
A very common mistake when differentiating implicit functions (where y is not explicitly solved for x) is forgetting to apply the Chain Rule to the 'y' terms. When you differentiate a term containing 'y' with respect to 'x', you must multiply the result by dy/dx. For example, the derivative of y² with respect to x is not just 2y; it is 2y · (dy/dx). To avoid this, always remember that you are differentiating with respect to x, and any other variable is treated as a function of x.
6. Why is logarithmic differentiation a useful technique for certain complex functions in RS Aggarwal Chapter 10?
Logarithmic differentiation is an essential technique, not just a shortcut. Its primary utility is for functions that are difficult or impossible to differentiate directly. It is most useful in two specific cases:
- When the function involves a variable raised to the power of another variable, such as xsin(x).
- When the function is a complex product or quotient of multiple functions, such as √((x-1)(x-2))/((x-3)(x-4)).
By taking the natural logarithm of both sides, you can use log properties to convert complex products, quotients, and exponents into simpler sums and differences, which are much easier to differentiate.
7. How are second-order derivatives approached in RS Aggarwal, and what do they represent conceptually?
In RS Aggarwal, second-order derivatives are introduced as the process of differentiating the first derivative of a function. If y = f(x), the first derivative is dy/dx, and the second-order derivative is found by differentiating dy/dx again with respect to x, denoted as d²y/dx² or f''(x).
Conceptually, the second-order derivative represents the rate of change of the rate of change. For instance, if the first derivative represents velocity, the second derivative represents acceleration. It is also used to determine the concavity of a curve and locate its points of inflection.
8. How does the concept of differentiability relate to the continuity of a function as per problems in RS Aggarwal?
The relationship is a fundamental theorem in calculus, often tested in RS Aggarwal problems. The key principle is: If a function is differentiable at a point, then it must be continuous at that point. However, the reverse is not always true. A function can be continuous at a point but not differentiable there. A classic example is the absolute value function, f(x) = |x|, which is continuous at x=0 but not differentiable at that point because of the sharp corner. Problems in the book often require you to check both conditions to fully analyse a function's behaviour.
