Differentiation Rules and Formulas: Overview For Class 11 & 12

Howdy Students!

You know, differentiation is an essential topic in calculus. It helps us understand how a function changes as its input changes (Instantaneous Rate of Change). In this article, we’ll break down the Differentiation Rules and Formulas you need to master, making complex concepts simple and easy to remember.

Moreover, it is very important to learn, understand and practice them while solving a number of problems, so that you’re well-prepared for the topics like Application of Derivatives and Integration later on.

By the end of today’s topic, you’ll have a clearer understanding of how to handle derivatives of various functions and their different forms.

1. Most Important Differentiation Rules

The foundation of differentiation lies in a few important rules that apply to different types of functions:

Sum and Difference Rule:

If you’re differentiating two functions added or subtracted together, the derivative is simply the sum or difference of their individual derivatives.

Product Rule:

For two functions being multiplied, you differentiate them using the product rule:

Quotient Rule:

For two functions being divided:

Chain Rule of Differentiation: The Most Important One

When differentiating a composite function like , the chain rule applies:

In my words, I would explain Chain Rule as, “Keep differentiating (and multiplying) all functions from the outermost to the innermost until you reach (x)!”

Here is an image (a quick reference) for all these important rules of differentiation —

These rules form the backbone of differentiation and are vital when working with more complex problems. We will see that in the upcoming topics.

---

2. Frequently Used Differentiation Formulas

Some functions and their similar forms are frequently required to be differentiated, therefore knowing their derivatives is essential.

Here are a few important ones:

Power Rule of Differentiation:

Exponential Functions:

○ Here, the condition on ‘a’ is to be positive and not equal to 1.

Logarithmic Functions:

Further,

where means natural logarithms

○ Again, ‘a’ has to be positive and can not be equal to 1 here.

If all these functions are part of a composite function, remember to apply the chain rule by replacing    , (or simply f for quick reference) and then differentiating accordingly. As you can see this in the image below —

Remember:

• Derivative of a constant function is always equal to zero because there is nothing which is going to change in it with respect to any other variable!
• But, if a constant is multiplied by some function, say ‘ f ’ then we just take it out of the differentiation and we get the derivative as, constant × (f’), e.g. differentiation of  2x² is 4x and that of 3x³ will be 9x².
• You will notice here that along with Differentiation Rules and Formulas, we must keep in mind, which type of function we are differentiating and ‘with respect to’ (w.r.t.) what function/variable. Because, that actually changes the flow of differentiation and we get the derivative accordingly.

---

3. Differentiation Rules and Formulas: Trigonometric Functions

When differentiating trigonometric functions, following are the basic derivatives:

• 
• 
• 
• 
• 
• 

*You can read csc x as cosec x only.

If the argument (i.e. the angle, in simple words!) of the trigonometric function is a more complex function like some , apply the chain rule as you can see below, e.g., derivative of sin(2x) is 2cos(2x), that of sin(x²) will be 2x cos(x²) and differentiation of {sin(cos x)} with respect to x will be “(–sin x)cos(cos x)” and likewise. Refer the following image —

Remember:

• You must have noticed that only the t-functions starting with ‘co-‘ i.e. cosine, cotangent and cosecant, have their derivatives with a minus sign.
• Learn them pairwise as there is a visible trend, (you can read the remarks in blue in the image above), for remembering and recollecting them later on easily. Similarly, you must have seen the trend and easy ways to learn the Trigonometric Identities, as I have explained in my previous article as well. 

---

4. Differentiation Rules and Formulas: Inverse Trigonometric Functions

Inverse trigonometric functions have their own set of rules as following:

• 
• 
• 

Similarly, the differentiation rules will apply on the other inverse t-functions also, as shown below —

Actually, these derivatives come in handy when solving more advanced problems in differentiation.

○ Here also, you’ll notice that the complementary ones have there derivatives with a negative sign just like we have it for T-Functions above.

---

5. Differentiating Exponential and Logarithmic Functions

Exponential and logarithmic functions are frequently used in higher-level calculus. For example, the properties of logarithms make it easier to differentiate complex functions involving products or quotients:

• 
• 

○ Using the properties of logarithms, you can simplify expressions before applying differentiation rules and formulas.

Understand and learn them as mentioned below along with practicing a good number of questions.

---

Practice Questions

1. Differetiate y = x²sin x
2. Find the derivative of ln(sin x)
3. What’s the derivative of arctan(3x)

Infact, you must try as many questions as you can on your own. Refer the above set of differentiation rules and formulas anytime. Should you require any help, do interact, I shall be more than happy to solve the problems.

Final Thoughts

Mastering Differentiation Rules and Formulas is crucial for solving calculus problems effectively. Whether you’re dealing with basic polynomials, trigonometric functions, or more complex logarithmic and exponential functions, understanding the rules will make differentiation much easier. Practice is key, so be sure to work on a variety of problems to strengthen your skills. Happy differentiating!

---