NCERT Solutions For Class 12 Maths Chapter 5 Exercise 5.4 Continuity and Differentiability

The ex 5.4 class 12 math portion is all about telling the difference between exponential and logarithmic functions, which is an important aspect of calculus. By solving these specific problems, you prepare yourself for more difficult math tasks. These answers show you how to use the chain rule and basic principles of e^x and log x to work with complicated derivatives step by step.

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NCERT Solutions For Class 12 Maths Chapter 5 Exercise 5.4

Solve The Following Questions NCERT Solutions For Class 12 Maths Chapter 5 Exercise 5.4 Continuity and Differentiability:

Question 1. Differentiate the following w.r.t. x:

Solution :

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1

Question 2. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.2

Question 3. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3

Question 4. Differentiate the following w.r.t. x: sin (tan–1 e ^-x ) 

Solution : Let, y = sin (tan–1 e ^-x ) By using the chain rule, we obtain

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5

Question 5. Differentiate the following w.r.t. x: log(cos e ^x ) 

Solution : Let y = log(cos e ^x ) By using the chain rule, we obtain

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6

Question 6.  Differentiate the following w.r.t. x:

Solution :

NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7

Question 7. Differentiate the following w.r.t. x:

Solution : Let y =

Question 8. Differentiate the following w.r.t. x: log(log x), x > 1 

Solution : Let y = log (log x),x > 1 By using the chain rule, we obtain

Question 9. Differentiate the following w.r.t. x:

Solution : Let y =By using the quotient rule, we obtain

Question 10. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

Solve The Following Questions.

Question 1. Differentiate the following w.r.t. x:

Solution :

Question 2. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

Question 3. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

Question 4. Differentiate the following w.r.t. x: sin (tan–1 e ^-x ) 

Solution : Let, y = sin (tan–1 e ^-x ) By using the chain rule, we obtain

Question 5. Differentiate the following w.r.t. x: log(cos e ^x ) 

Solution : Let y = log(cos e ^x ) By using the chain rule, we obtain

Question 6.  Differentiate the following w.r.t. x:

Solution :

Question 7. Differentiate the following w.r.t. x:

Solution : Let y =

Question 8. Differentiate the following w.r.t. x: log(log x), x > 1 

Solution : Let y = log (log x),x > 1 By using the chain rule, we obtain

Question 9. Differentiate the following w.r.t. x:

Solution : Let y =By using the quotient rule, we obtain

Question 10. Differentiate the following w.r.t. x:

Solution : Let y =By using the chain rule, we obtain

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Learn How to do Derivatives using Ex 5.4 Class 12 Maths NCERT Solutions

Calculus can seem like a mountain, but if you split it down, it's easier to climb. Exercise 5.4 in Chapter 5 connects simple differentiation to the more complicated domain of transcendental functions. In instance 5.4 class 12 maths, you won't simply be looking for an answer; you'll also be learning how the rate of change works for natural logarithms and Euler's number.

We put together these ex 5.4 class 12 math solutions so that you may understand the logic behind the formulas instead of just memorizing them. The goal is still the same, whether you're comparing these to :eg 5.4 class 12 maths rd sharma questions or keeping exclusively to the NCERT curriculum: correctness and speed. The best thing you can do is practice. If you've already done ex 5.4 class 12 maths 1, you know that the chain rule is the "secret sauce" for quickly solving these problems.

Learning how to tell the difference between exponential and logarithmic functions 

The core of ex 5.4 class 12 maths revolves around two primary functions. First, we have the exponential function, y = e^x. A unique property of this function is that its derivative is the function itself. This makes it a favorite in calculus exams. However, when the exponent is a function of x, like e^sin x, you must apply the chain rule.

Then, we have the logarithmic function, y = log x. The derivative of log x with respect to x is 1/x. In this exercise, we often see these functions combined with trigonometric or algebraic terms. For instance, differentiating log(cos e^x) requires you to peel back the layers of the function one by one. You start from the outermost log and work your way into the innermost exponential term.

Step-by-Step Breakdown of Problems in Ex 5.4

You need a systematic way to do well on your boards. Let's talk about how to deal with the different kinds of questions in the ex 5.4 class 12 mathematics ncert solutions.

Quotients of Functions: Some issues have fractions in them, as e^x / sin x. Use the quotient rule here. Keep in mind the formula: (u/v)' = (vu' - uv') / v^2. It's an important tool you need.

Functions with a Composite Exponential: Don't worry when you see e^(x^2) or e^cos x. First, find the derivative of the "e" part. Then, multiply it by the derivative of the exponent.

Layers with logarithms: It's usual to see functions like log(log x). First, treat the inner log x as one variable. The derivative is (1 / log x) * (1/x).

Common Problems and Helpful Tips 

Students usually make mistakes when they forget to check the domain. Logarithms can only be used with real numbers that are positive. The goal of the exercise is to learn how to differentiate, however knowing these things will help you not make mistakes. Another mistake that happens a lot is mixing up the derivative of a^x with e^x. Keep in mind that the derivative always has a log a term for each base a.

We think you should do these things more than once. Before you undertake the task, look at the examples in the NCERT book. If the NCERT problems are too easy for you, you can make them harder by looking at ex 5.4 class 12 maths rd sharma. This is helpful for tests like the JEE that are competitive.

Why Every Step Matters

Examiners want clear answers in the board exams. You could lose points even if your final answer is right if you skip steps in the chain rule. Write down the formula that you are employing. If you're utilizing the quotient rule, make sure to say what u and v are. This methodical way of doing things lowers the risk of making foolish mistakes in math. It also makes it easier for you to examine your work again in the last few minutes of the test.

The more you do something, the better you get at it. Keep going until you can look at a function like square root of (e^square root of x) and mentally map out the three steps of differentiation you need to take to solve it. You can only get this level of knowledge by seeing the same 5.4 class 12 math patterns over and over again.

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Frequently Asked Questions (FAQs)

Q1: What is the major topic of math class 12 ex 5.4?

The main goal of this exercise is to learn how to tell the difference between logarithmic functions (log x) and exponential functions (e^x). It shows students how to use the chain rule, product rule, and quotient rule on various kinds of transcendental functions.

Q2: How does the chain rule work with the answers to ex 5.4 class 12 maths ncert?

When the argument of an exponential or logarithmic function is made up of more than one part, the chain rule is applied. In d/dx (log(sin x)), for instance, you first find the derivative of the log function (1/sin x) and then multiply it by the derivative of sin x (cos x).

Q3: Do the answers to the 12th grade math questions in instance 5.4 help you become ready for the board exam?

You can get a rudimentary understanding from the NCERT solutions, however JEE requires you to solve harder problems. We suggest that you study using ex 5.4 class 12 maths rd sharma or PW's specific JEE modules to aid you with the difficult questions.

Q4: Where can I find more practice questions similar to ex 5.4 class 12 maths?

 You can find an extensive collection of practice problems and solved examples in the PW Store. Their Class 12 Mathematics question banks offer variety beyond the standard NCERT exercise to help you build confidence for all exam formats.

Q5: Why is it vital to understand how to solve class 12 maths 1 problems?

These problems show you functions that change quickly. For subjects like Integration and Differential Equations, which are very important for the final board exams, it's important to know how to find their derivatives.