How to Solve Differentiation Questions Using Rules Formulas and Solved 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 and Detailed Solutions
1. What is differentiation in calculus?
Differentiation is the process of finding the derivative, which measures the rate of change of a function with respect to a variable. In simple terms, it tells us how fast one quantity changes when another quantity changes.
- If y = f(x), then its derivative is written as dy/dx or f'(x).
- Geometrically, differentiation gives the slope of the tangent line to a curve at a point.
- It is widely used in physics (velocity, acceleration), economics (marginal cost), and optimisation problems.
2. What is the formula for the derivative from first principles?
The derivative from first principles is defined as f'(x) = lim(h→0) [f(x+h) − f(x)] / h. This limit measures the instantaneous rate of change of the function.
- Start with the difference quotient: (f(x+h) − f(x)) / h.
- Expand and simplify the expression.
- Take the limit as h → 0.
3. What is the power rule in differentiation?
The power rule states that if f(x) = xⁿ, then its derivative is f'(x) = n·xⁿ⁻¹. This rule is one of the most commonly used differentiation formulas.
- Example: If f(x) = x⁵, then f'(x) = 5x⁴.
- Example: If f(x) = 3x², then f'(x) = 6x.
4. How do you differentiate a constant?
The derivative of a constant is always 0. A constant does not change with respect to x, so its rate of change is zero.
- If f(x) = 7, then f'(x) = 0.
- If f(x) = −3, then f'(x) = 0.
5. What is the product rule in differentiation?
The product rule is used to differentiate the product of two functions and is given by (uv)' = u'v + uv'. It ensures both functions are properly differentiated.
- Let y = x²·sinx.
- Then y' = (2x)(sinx) + (x²)(cosx).
6. What is the quotient rule in differentiation?
The quotient rule is used when dividing two functions and is given by (u/v)' = (u'v − uv') / v². It helps differentiate rational functions correctly.
- Let y = x / (x² + 1).
- Then y' = [(1)(x²+1) − x(2x)] / (x²+1)².
7. What is the chain rule in differentiation?
The chain rule is used to differentiate composite functions and is written as d/dx[f(g(x))] = f'(g(x))·g'(x). It accounts for the rate of change of both inner and outer functions.
- Example: If y = (3x² + 1)⁴,
- Outer derivative: 4(3x²+1)³
- Inner derivative: 6x
- Final answer: 24x(3x²+1)³
8. How do you find the derivative of trigonometric functions?
The derivatives of basic trigonometric functions follow standard formulas such as d/dx(sin x) = cos x and d/dx(cos x) = −sin x.
- d/dx(tan x) = sec²x
- d/dx(sec x) = secx·tanx
- d/dx(cosec x) = −cosecx·cotx
- d/dx(cot x) = −cosec²x
9. What is the geometric meaning of a derivative?
The derivative at a point represents the slope of the tangent line to the curve at that point. It shows how steep the curve is and whether it is increasing or decreasing.
- If f'(x) > 0, the function is increasing.
- If f'(x) < 0, the function is decreasing.
- If f'(x) = 0, the point may be a maximum or minimum.
10. What are common mistakes in differentiation?
Common mistakes in differentiation include misapplying rules and forgetting to differentiate inner functions. Avoiding these errors improves accuracy in calculus problems.
- Forgetting the chain rule for composite functions.
- Incorrect signs in trigonometric derivatives (e.g., derivative of cos x is −sin x).
- Not squaring the denominator in the quotient rule.
- Differentiating constants incorrectly.
