

What are the basic rules of differentiation in maths?
The concept of Differentiation Formulas plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding these formulas helps students solve problems related to rates of change, tangents, and optimization in calculus.
What Is Differentiation Formula?
A Differentiation Formula is a mathematical expression used to find the derivative, or the instantaneous rate of change, of a function with respect to its independent variable. You’ll find this concept applied in areas such as calculating velocity in Physics, rate of reaction in Chemistry, and profit changes in Business Studies.
Key Formula for Differentiation
Here’s the standard differentiation formula for the power of x: \( \frac{d}{dx}(x^n) = n x^{n-1} \). Below, you’ll see the most important formulas for polynomials, trigonometric, exponential, and logarithmic functions – essential for exams and competitive tests like JEE and CBSE.
Function | Derivative |
---|---|
k (constant) | 0 |
xn | n xn-1 |
sin x | cos x |
cos x | -sin x |
tan x | sec2 x |
ex | ex |
ln(x) | 1/x |
Common Differentiation Rules
Some functions require special rules for differentiation:
- Sum / Difference Rule: The derivative of [f(x)±g(x)] is [f'(x)±g'(x)]
- Product Rule: The derivative of [f(x)·g(x)] is [f'(x)·g(x) + f(x)·g'(x)]
- Quotient Rule: The derivative of [f(x)/g(x)] is [f'(x)·g(x) – f(x)·g'(x)] / [g(x)]²
- Chain Rule: The derivative of f(g(x)) is f'(g(x))·g'(x)
Step-by-Step Illustration
- Differentiate \( y = 4x^2 + x - 4 \) with respect to x
- Add up each term:
\( \frac{dy}{dx} = 8x + 1 \)
\( \frac{d}{dx}(4x^2) = 4·2x = 8x \)
\( \frac{d}{dx}(x) = 1 \)
\( \frac{d}{dx}(-4) = 0 \)
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for monomials: To differentiate \( ax^n \), simply multiply exponent by the coefficient and reduce the exponent by 1.
Example: \( \frac{d}{dx}(7x^4) = 4 × 7x^{3} = 28x^3 \)
Tricks like this are practical for quick MCQ solving in Olympiads and JEE. Vedantu’s live classes cover more tips and calculator hacks for speed and efficiency.
Try These Yourself
- Differentiate \( f(x) = 5x^3 \)
- Find the derivative of \( \sin x \cdot e^x \)
- Differentiate \( y = \frac{x^2+1}{x} \)
- Apply the chain rule to \( y = \ln(2x^2+3) \)
Frequent Errors and Misunderstandings
- Forgetting to use the product or quotient rule when multiplying or dividing two functions.
- Missing the chain rule in nested functions like \( \sin(3x^2) \).
- Sign errors when differentiating trigonometric or negative powers.
Relation to Other Concepts
The idea of differentiation formulas connects closely with other calculus topics such as Integration Formula and Limits and Derivatives. Mastering these basics is important for learning advanced concepts like optimization, analytic geometry, and curve sketching.
Classroom Tip
A quick way to remember differentiation rules: “Sum stays, Product gets split, Quotient gets a minus, Chain follows the link.” Vedantu’s expert teachers often use charts and colored cues to make formulas easier to recall during live sessions.
We explored Differentiation Formulas—from definition, formula, stepwise examples, errors, and links to other calculus topics. Keep practicing with Vedantu’s differentiation formula page to build confidence and accuracy in problems involving derivatives.
Additional Learning Resources
FAQs on Differentiation Formulas – Rules, Table, and Examples
1. What is differentiation in Class 12 Maths?
In Class 12 Maths, differentiation is a fundamental concept in calculus used to find the instantaneous rate of change of a function with respect to one of its variables. Geometrically, the derivative of a function at a point gives the slope of the tangent line to the function's graph at that exact point. It is a crucial tool for analysing the behaviour of functions.
2. What are the fundamental rules of differentiation every Class 12 student must know?
For the CBSE Class 12 syllabus, every student must master these fundamental rules:
- Power Rule: Used for differentiating functions of the form xn. The rule is d/dx(xn) = nxn-1.
- Sum/Difference Rule: To differentiate a sum or difference of two functions, you differentiate each function separately: d/dx[f(x) ± g(x)] = f'(x) ± g'(x).
- Product Rule: For differentiating the product of two functions: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).
- Quotient Rule: For differentiating a function that is a ratio of two functions: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]2.
- Chain Rule: Essential for differentiating composite functions (a function within a function): d/dx[f(g(x))] = f'(g(x)) ⋅ g'(x).
3. How do you decide which differentiation rule to apply to a problem?
To select the correct rule, analyse the structure of the function:
- If the function is a simple power like x5, use the Power Rule.
- If the function involves terms being added or subtracted, like sin(x) + x2, use the Sum/Difference Rule on each term.
- If two functions are multiplied, such as ex⋅log(x), apply the Product Rule.
- If one function is divided by another, like tan(x)/x, use the Quotient Rule.
- If you see a function inside another function, like cos(x3), you must use the Chain Rule.
4. What is the difference between differentiation and integration?
Differentiation and integration are inverse operations in calculus, as stated by the Fundamental Theorem of Calculus. The key difference is their purpose:
- Differentiation finds the instantaneous rate of change or the slope of a function. It breaks a function down to analyse its change at a point.
- Integration finds the accumulation of quantities, which is geometrically represented as the area under the curve of a function. It builds a function up from its rate of change.
5. What are some real-world examples of differentiation?
Differentiation is used to model and solve problems involving rates of change across many fields:
- In Physics, it's used to calculate velocity (rate of change of displacement) and acceleration (rate of change of velocity).
- In Economics, it helps determine marginal cost and marginal revenue, which are crucial for finding the optimal production level to maximise profit.
- In Engineering, it is used for optimisation problems, like finding the dimensions that minimise material usage for a container of a given volume.
- In Biology, it can model the rate of growth of a population of bacteria or the rate at which a medicine's concentration in the body changes.
6. How is the concept of a derivative connected to limits and continuity?
The connection is foundational to calculus. A function must be continuous at a point to be differentiable at that point, but continuity alone does not guarantee differentiability. The formal definition of a derivative is expressed as a limit: f'(x) = lim (as h→0) of [f(x+h) - f(x)]/h. If this limit does not exist, the function is not differentiable at that point. This happens at sharp corners, cusps, or vertical tangents, even if the function is continuous there.
7. What common mistakes should be avoided when applying the product and quotient rules?
Students often make specific, preventable errors with these rules. For the Product Rule, a common mistake is to simply multiply the derivatives of the two functions (f'(x)g'(x)) instead of using the correct formula f'(x)g(x) + f(x)g'(x). For the Quotient Rule, the most frequent error is mixing up the order of terms in the numerator; it must be f'(x)g(x) - f(x)g'(x). A sign error here will lead to a completely incorrect answer.
8. Why is a function not differentiable at a sharp corner or a cusp?
A function is not differentiable at a sharp corner or cusp because the concept of a unique tangent line breaks down at that point. The derivative represents the slope of the tangent. At a sharp point, the slope as you approach from the left is different from the slope as you approach from the right. Since the left-hand and right-hand limits of the derivative are not equal, the overall limit does not exist, and therefore, the function is not differentiable there.
9. What is implicit differentiation and in what situations is it necessary?
Implicit differentiation is a technique used when you have a relationship between x and y that is difficult or impossible to solve for y explicitly as a function of x. It is necessary for equations like x² + y² = 25 or x³ + y³ = 3xy. To use it, you differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule whenever you differentiate a term containing y. Finally, you algebraically solve for dy/dx.
10. Why is the chain rule considered one of the most crucial rules in differentiation?
The chain rule is crucial because most functions encountered in real-world applications are not simple polynomials but are composite functions—functions made by combining other functions. Without the chain rule, we could not differentiate expressions like sin(2x), e-x², or ln(x² + 1). It allows us to break down complex, layered functions into manageable parts, differentiate them individually, and then multiply the results to find the overall rate of change.

















