How to Solve Differentiation Questions in Maths – Rules, Steps & Examples
The concept of differentiation questions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Differentiation is a fundamental topic in calculus, and being able to solve different types of differentiation questions is essential for success in Class 11 and 12, as well as competitive exams like JEE and NEET.
What Is Differentiation in Maths?
A differentiation question asks you to find the derivative of a function — that is, to measure how a function changes as its input changes. You’ll find this concept applied in areas such as physics (to find speed or acceleration), economics (to find marginal cost), and biology (to model rates of change in populations). Differentiation is also essential for graphing curves, finding minimum and maximum points, and solving real-world optimisation problems.
Key Formula and Rules for Differentiation Questions
Here are the standard and most important formulas for solving differentiation questions:
| Function f(x) | Derivative f'(x) | Rule Name |
|---|---|---|
| k (constant) | 0 | Constant Rule |
| xn | n xn-1 | Power Rule |
| ex | ex | Exponential Rule |
| ln(x) | 1/x | Logarithm Rule |
| sin(x), cos(x), tan(x) | cos(x), -sin(x), sec2(x) | Trig Rules |
| u(x)×v(x) | u'v + uv' | Product Rule |
| u(x)/v(x) | (u'v - uv')/v2 | Quotient Rule |
| f(g(x)) | f'(g(x)) × g'(x) | Chain Rule |
These rules make different types of differentiation questions easy to solve. Practising with these will help you avoid missing important steps—especially under exam time pressure!
How to Approach Differentiation Questions – Step-by-Step
Follow these steps to solve any differentiation question:
1. Identify the function type you need to differentiate (polynomial, product, quotient, composite function, etc.).2. Choose the right rule (from the table above) — Power, Product, Quotient, or Chain.
3. Write each differentiation step clearly. Use brackets to avoid calculation errors.
4. Simplify your answer at every step – combine like terms and reduce fractions.
5. Double-check with the original question — did you differentiate with respect to the correct variable?
Remember, writing out each step will save you marks in board and competitive exams!
Solved Differentiation Questions — From Easy to Advanced
Let’s see how to solve different types of differentiation questions with clear steps.
Example 1: Differentiate x5 with respect to x.
1. Let y = x52. Using the Power Rule: d(xn)/dx = n xn-1
3. dy/dx = 5x4
**Final Answer:** 5x4
Example 2: Differentiate 10x2 with respect to x.
1. Let y = 10x22. dy/dx = 10 × d(x2)/dx = 10 × 2x = 20x
**Final Answer:** 20x
Example 3: Differentiate (3x – 7)(x2 + 4x) with respect to x.
1. Let u = 3x – 7, v = x2 + 4x2. Use Product Rule:
d(uv)/dx = u'v + uv'
3. u' = 3, v' = 2x + 4
4. dy/dx = 3(x2 + 4x) + (3x – 7)(2x + 4)
5. Expand and simplify as needed.
**Final Answer:** 3x2 + 12x + (6x2 + 12x – 14x – 28) = 9x2 + 10x – 28
Example 4: Differentiate sin(3x+5) with respect to x.
1. Let y = sin(3x+5)2. Use Chain Rule:
dy/dx = cos(3x+5) × d(3x+5)/dx
3. d(3x+5)/dx = 3
**Final Answer:** 3cos(3x+5)
Example 5: Differentiate (a2 – x2)/(a2 + x2)½ with respect to x.
1. Let y = [(a2 – x2)/(a2 + x2)]½2. Take log both sides, simplify using logarithm rules.
3. Use Chain Rule, Quotient Rule, and return to y at the end.
**This problem uses all main rules—great exam practice!**
Practise These Differentiation Questions Yourself
- Differentiate tan2x with respect to x.
- Differentiate x3ex with respect to x.
- Find d/dx [loge(x2 + 1)]
- Differentiate cos–1x with respect to x.
- Differentiate e3x × sin x with respect to x.
Frequent Errors and Exam Tips for Differentiation Questions
- Forgetting to use Chain Rule on composite functions.
- Leaving out constants or coefficients in answers.
- Not using parentheses, causing calculation errors.
- Mixing up the Product and Quotient Rules.
- Forgetting d(constant)/dx = 0.
To avoid these mistakes, always write every differentiation step neatly. During exams, underline key steps to reduce careless errors. Vedantu’s expert faculty recommend practising at least 20 mixed differentiation questions before any test.
Connections to Other Maths Concepts
Understanding differentiation questions helps in mastering related calculus topics such as integration, applications of derivatives, curve sketching, and optimisation. For deeper learning, see our page on differentiation formulas and explore chain rule problems for more challenging practice.
Classroom Tip
A quick way to decide which rule for differentiation to use is by looking at the structure of the function — single term (power rule), two multiplied (product rule), two divided (quotient rule), or function inside a function (chain rule). Vedantu’s teachers often teach students to write “Type? Rule!” in the margin to remind themselves which formula to pick.
We explored differentiation questions—from definition and rules to solved examples, mistakes, and connections to other subjects. With regular practice and clear stepwise solving, you can become confident in all types of differentiation problems. Continue using Vedantu’s solved worksheets and live classes to strengthen your calculus foundation!
Further Exploration and Internal Links
- Differentiation Formula – For all important rules, with examples
- Derivatives – Deeper dive into theory and advanced differentiability
- Partial Derivative – For higher-level competitive and multivariable questions
- Limits and Derivatives – Refresh basic calculus concepts for Class 11 and 12
FAQs on Differentiation Questions: Practice Problems with Solutions
1. What is differentiation in mathematics?
Differentiation is a fundamental concept in calculus. It's the process of finding the derivative of a function. The derivative measures the instantaneous rate of change of a function with respect to one of its variables. In simpler terms, it tells us how much a function's output changes when its input changes by a tiny amount. This is crucial for understanding concepts like slopes of curves, velocity, acceleration, and optimization problems.
2. What are the basic rules of differentiation?
Several rules simplify the process of finding derivatives. Key rules include the power rule, the product rule, the quotient rule, and the chain rule. Each rule provides a shortcut for differentiating specific types of functions. For example, the power rule states that the derivative of xn is nxn-1. Mastering these rules is essential for solving differentiation questions effectively.
3. How do I solve differentiation problems step-by-step?
Solving differentiation problems involves a systematic approach: 1. Identify the function you need to differentiate. 2. Choose the appropriate rule based on the function's structure (power rule, product rule, etc.). 3. Apply the rule carefully, differentiating each part of the function according to the rule. 4. Simplify the resulting expression. Remember to use appropriate algebraic techniques to obtain the simplest form of the derivative.
4. What are some common mistakes to avoid when differentiating?
Common errors include forgetting to apply the chain rule correctly, making mistakes with signs when using the product or quotient rule, and errors in simplifying the resulting expressions. Careful attention to detail and consistent practice will reduce errors. It is also helpful to check your work by using various methods or tools.
5. What are the applications of differentiation?
Differentiation has widespread applications in various fields. It is used to determine the maximum or minimum values of functions (optimization), find rates of change in physics and engineering (velocity, acceleration), analyze curves and tangents in geometry, and model real-world phenomena in areas such as economics and biology.
6. How is differentiation related to integration?
Differentiation and integration are inverse operations. Differentiation finds the rate of change of a function, while integration finds the area under a curve, which is also the antiderivative of the function. Understanding this relationship is crucial for advanced calculus concepts.
7. What is the difference between a derivative and a differential?
The derivative is the instantaneous rate of change of a function, often represented as dy/dx or f'(x). The differential, represented as dy, approximates the change in the function's value (Δy) when the independent variable changes by a small amount (dx).
8. How can I improve my speed in solving differentiation problems?
Practice is key! Consistent practice with a variety of problems will improve both speed and accuracy. Focus on understanding the underlying concepts, mastering the rules, and practicing simplification techniques. Working through past papers can be beneficial too.
9. Where can I find more practice problems on differentiation?
Numerous resources are available for practicing differentiation. Textbooks, online platforms (such as Vedantu), and educational websites provide practice problems with solutions. Past exam papers are also an excellent source of questions to work through.
10. What is implicit differentiation?
Implicit differentiation is a technique used to find the derivative of a function that is not explicitly expressed as y = f(x). Instead, the function is defined implicitly by an equation relating x and y. The method involves differentiating both sides of the equation with respect to x, using the chain rule where necessary, and then solving for dy/dx.
11. What is logarithmic differentiation?
Logarithmic differentiation is a technique that simplifies the differentiation of complicated functions, especially those involving products, quotients, and powers of functions. It involves taking the natural logarithm of both sides of the equation, applying logarithmic properties to simplify the expression, and then differentiating implicitly.
12. How can I verify if my solution to a differentiation problem is correct?
Several methods can verify your solution. You can compare your answer to solutions provided in textbooks or online resources. You could also use graphical methods to check if the derivative matches the slope of the function's graph at various points. Some advanced calculators or software can also help you verify your answers.
