Differentiation – Basic Derivatives for JEE Main | Formulas, Rules & Solved Questions

Differentiation – Basic Derivatives | JEE Main Maths

Differentiation is one of the most important foundations of Calculus for JEE Mathematics. It tells us how rapidly a quantity changes with respect to another quantity. Once the basic derivative formulas and rules become automatic, many JEE Main questions can be solved quickly.

In this chapter, we will cover the essential derivative rules including the power rule, sum and difference rule, trigonometric derivatives, exponential and logarithmic derivatives, inverse trigonometric derivatives, product rule, quotient rule and chain rule.

The most important approach is:

Identify the Function → Identify the Rule → Differentiate → Simplify

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1. What is Differentiation?

Differentiation tells us how fast a function changes with respect to x.

If:

y = f(x)

then its derivative with respect to x is written as:

dy/dx = f′(x)

Geometrical Meaning

Geometrically, the derivative represents the slope of the tangent to the curve at a point.

For example:

y = x²

Differentiating:

dy/dx = 2x

Therefore, at any value of x, 2x gives the slope of the tangent to the curve y = x².

Differentiation is therefore useful not only for calculating derivatives but also for understanding the behavior and rate of change of functions.

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2. Basic Differentiation Rules

Rule 1: Derivative of a Constant

The derivative of any constant is zero.

d/dx(c) = 0

Example

d/dx(7) = 0

Why? A constant does not change when x changes. Therefore, its rate of change is zero.

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Rule 2: Power Rule

The power rule is one of the most frequently used differentiation rules.

d/dx(xⁿ) = nxⁿ⁻¹

Examples

d/dx(x²) = 2x

d/dx(x⁵) = 5x⁴

d/dx(x¹⁰) = 10x⁹

The shortcut is simple:

Power comes down × Power decreases by 1

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Rule 3: Constant Multiple Rule

If a function is multiplied by a constant, the constant can be kept outside the differentiation.

d/dx[c f(x)] = c f′(x)

Example

Find:

d/dx(5x³)

Using the power rule:

5 × 3x² = 15x²

Therefore:

d/dx(5x³) = 15x²

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3. Sum and Difference Rule

If two or more functions are added or subtracted, differentiate each term separately.

d/dx[f(x) ± g(x)] = f′(x) ± g′(x)

Example

Let:

y = x³ + 4x² − 7x + 5

Differentiate each term:

dy/dx = 3x² + 8x − 7

The derivative of the constant 5 is zero, so it disappears.

dy/dx = 3x² + 8x − 7

JEE Tip: For a polynomial containing only sums and differences, differentiate each term separately. There is no need for product or quotient rules unless the structure actually requires them.

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4. Standard Derivatives – Trigonometric Functions

These derivatives should be memorized because they appear repeatedly in JEE questions.

Function	Derivative
sin x	cos x
cos x	−sin x
tan x	sec²x
cot x	−cosec²x
sec x	sec x tan x
cosec x	−cosec x cot x

🔥 JEE Memory Shortcut

sin → cos
cos → −sin
tan → sec²
cot → −cosec²

For sec and cosec:

sec → sec tan

cosec → −cosec cot

The negative sign in the derivatives of cos x, cot x and cosec x is a common source of mistakes.

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5. Exponential and Logarithmic Derivatives

Exponential Functions

The most important exponential derivative is:

d/dx(e^x) = e^x

The function e^x is special because its derivative is itself.

For a general positive base a:

d/dx(a^x) = a^x ln a

Logarithmic Functions

For the natural logarithm:

d/dx(ln x) = 1/x

For logarithm to base a:

d/dx(log_ax) = 1/(x ln a)

Example 1

Find:

d/dx(e^3x)

The outer function is exponential, while the exponent contains 3x. Therefore, the chain rule is required.

Result:

3e^3x

Example 2

Find:

d/dx(ln 5x)

Using the derivative of the logarithmic function:

d/dx(ln 5x) = 1/x

Here, the constant factor inside the logarithm does not change the final derivative in this form.

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6. Derivatives of Inverse Trigonometric Functions

Inverse trigonometric derivatives are very important for JEE Main and should be memorized carefully.

Function	Derivative
sin−1x	1/√(1−x²)
cos−1x	−1/√(1−x²)
tan−1x	1/(1+x²)
cot−1x	−1/(1+x²)
sec−1x	1/(|x|√(x²−1))
cosec−1x	−1/(|x|√(x²−1))

Most Important for JEE

Two formulas that you should recognize instantly are:

d/dx(sin⁻¹x) = 1/√(1−x²)

d/dx(tan⁻¹x) = 1/(1+x²)

Important: The notation sin⁻¹x here means inverse sine, not 1/sin x.

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7. Product Rule

When two functions are multiplied together, the ordinary power rule is not enough. We use the product rule.

If:

y = u × v

then:

dy/dx = u(dv/dx) + v(du/dx)

or:

(uv)′ = u′v + uv′

Example

Let:

y = x² sin x

Take:

u = x²

v = sin x

Then:

u′ = 2x

v′ = cos x

Using product rule:

dy/dx = x² cos x + 2x sin x

Therefore:

dy/dx = x²cos x + 2xsin x

Memory Trick

First × derivative of Second + Second × derivative of First

Never forget the plus sign in the product rule.

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8. Quotient Rule

When one function is divided by another, use the quotient rule.

If:

y = u/v

then:

dy/dx = [v(du/dx) − u(dv/dx)]/v²

Memory Trick

Bottom × d(Top) − Top × d(Bottom)

————————————————————————

Bottom²

Example

Let:

y = x²/(x+1)

Here:

u = x²

v = x+1

Therefore:

u′ = 2x

v′ = 1

Using quotient rule:

dy/dx = [(x+1)(2x) − x²(1)]/(x+1)²

Simplifying:

= [2x² + 2x − x²]/(x+1)²

Therefore:

dy/dx = (x² + 2x)/(x+1)²

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9. Chain Rule

The chain rule is used when one function is inside another function.

If:

y = f(g(x))

then:

dy/dx = f′(g(x)) × g′(x)

In simple words:

Outer Derivative × Inner Derivative

Example

Consider:

y = (3x² + 1)⁵

Step 1 – Identify the Outer Function

The outer function is the fifth power.

Its derivative is:

5(3x²+1)⁴

Step 2 – Differentiate the Inner Function

The inner function is:

3x² + 1

Its derivative is:

6x

Step 3 – Multiply

Therefore:

dy/dx = 5(3x²+1)⁴ × 6x

Hence:

dy/dx = 30x(3x²+1)⁴

🔥 Chain Rule Shortcut

Outer derivative × Inner derivative

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10. Mixed Basic Differentiation Examples

Example 1 – Polynomial

Given:

y = 3x⁴ − 5x² + 7x − 9

Differentiate each term:

dy/dx = 12x³ − 10x + 7

The constant −9 disappears because its derivative is zero.

Answer: 12x³ − 10x + 7

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Example 2 – Chain Rule

Given:

y = sin(x²)

Outer function = sin

Inner function = x²

Therefore:

dy/dx = cos(x²) × 2x

Hence:

dy/dx = 2x cos(x²)

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Example 3 – Product Rule

Given:

y = x²e^x

Using product rule:

dy/dx = x²e^x + 2xe^x

Taking e^x common:

dy/dx = e^x(x² + 2x)

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11. JEE Main-Level Question

Question

If:

y = x³ + 3x² − 5x + 7

find dy/dx at x = 2.

Step 1 – Differentiate

Using the power rule:

dy/dx = 3x² + 6x − 5

Step 2 – Put x = 2

dy/dx |_x=2 = 3(2)² + 6(2) − 5

= 3(4) + 12 − 5

= 12 + 12 − 5

= 19

Final Answer = 19

This is a very common JEE pattern: differentiate first, substitute later.

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12. Differentiate First, Substitute Later

When a question asks for the derivative at a particular value of x, it is usually easiest to:

1. Find the derivative function.
2. Then substitute the given value of x.

For example, if:

f(x) = x³ + 3x² − 5x + 7

first find:

f′(x) = 3x² + 6x − 5

and only then put x = 2.

This keeps the calculation organized and reduces mistakes.

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13. Common Mistakes in Differentiation

❌ Mistake 1: Forgetting the Negative Sign in cos x

Remember:

d/dx(cos x) = −sin x

Not +sin x.

❌ Mistake 2: Forgetting the Inner Derivative

For:

y = sin(x²)

the answer is not simply cos(x²).

You must multiply by the derivative of x²:

2x cos(x²)

❌ Mistake 3: Wrong Order in Quotient Rule

Remember:

Bottom × d(Top) − Top × d(Bottom)

Do not reverse the subtraction.

❌ Mistake 4: Derivative of a Constant

The derivative of a constant is:

0

For example:

d/dx(7) = 0

It is not 1.

❌ Mistake 5: Using Power Rule Directly on a Product

For something such as:

x² sin x

you cannot differentiate the two factors independently and multiply their derivatives.

You must use the product rule.

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14. How to Identify Which Rule to Use?

Function Structure	Rule
xn	Power Rule
f(x) + g(x)	Sum Rule
f(x) − g(x)	Difference Rule
c f(x)	Constant Multiple Rule
u × v	Product Rule
u/v	Quotient Rule
f(g(x))	Chain Rule
sin x, cos x, tan x...	Standard Trigonometric Derivative
ex, ax	Exponential Derivative
ln x, logax	Logarithmic Derivative

Many JEE questions become easy once you correctly identify the structure of the function.

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15. JEE Main Quick-Solving Strategy

Before differentiating, look at the structure of the function.

Ask yourself:

1. Is it a simple power? → Power Rule
2. Is it a sum or difference? → Differentiate term by term
3. Is a constant multiplying the function? → Keep the constant outside
4. Are two functions multiplied? → Product Rule
5. Are two functions divided? → Quotient Rule
6. Is one function inside another? → Chain Rule
7. Is it a standard trigonometric, exponential or logarithmic function? → Use the standard derivative

The key is to recognize the structure before starting the calculation.

🔥 Basic Function → Identify Rule → Differentiate → Simplify

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16. Complete Basic Derivative Formula Sheet

Function	Derivative
c	0
xn	nxn−1
sin x	cos x
cos x	−sin x
tan x	sec²x
cot x	−cosec²x
sec x	sec x tan x
cosec x	−cosec x cot x
ex	ex
ax	ax ln a
ln x	1/x
logax	1/(x ln a)
sin−1x	1/√(1−x²)
cos−1x	−1/√(1−x²)
tan−1x	1/(1+x²)
cot−1x	−1/(1+x²)
sec−1x	1/(|x|√(x²−1))
cosec−1x	−1/(|x|√(x²−1))

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17. Product, Quotient and Chain Rule – Quick Sheet

Product Rule

(uv)′ = u′v + uv′

Quotient Rule

(u/v)′ = (vu′ − uv′)/v²

Chain Rule

[f(g(x))]′ = f′(g(x))g′(x)

🔥 Product → Multiply Structure
Quotient → Bottom dTop − Top dBottom
Chain → Outer × Inner

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18. One-Minute Revision

Basic Rules

d/dx(c) = 0

d/dx(xⁿ) = nxⁿ⁻¹

d/dx[cf(x)] = cf′(x)

d/dx[f ± g] = f′ ± g′

Trigonometric

d/dx(sin x) = cos x

d/dx(cos x) = −sin x

d/dx(tan x) = sec²x

d/dx(cot x) = −cosec²x

Exponential & Logarithmic

d/dx(e^x) = e^x

d/dx(a^x) = a^xln a

d/dx(ln x) = 1/x

d/dx(log_ax) = 1/(x ln a)

Important Rules

Product → u′v + uv′

Quotient → (vu′ − uv′)/v²

Chain → Outer derivative × Inner derivative

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19. Final Revision Box

🔥 DIFFERENTIATION – MUST REMEMBER

1. Constant: d/dx(c) = 0

2. Power: d/dx(xⁿ) = nxⁿ⁻¹

3. Sin: cos x

4. Cos: −sin x

5. Tan: sec²x

6. Cot: −cosec²x

7. e^x: e^x

8. ln x: 1/x

9. Product: u′v + uv′

10. Quotient: (vu′ − uv′)/v²

11. Chain: Outer × Inner

12. sin⁻¹x: 1/√(1−x²)

13. tan⁻¹x: 1/(1+x²)

🔥 Golden Rule:
Basic Function → Identify Rule → Differentiate → Simplify

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20. Practice Questions

Try these questions yourself before checking the formulas again.

1. Find the derivative of y = x⁷.
2. Find the derivative of y = 5x⁴ − 3x² + 7x − 2.
3. Differentiate sin x + cos x.
4. Differentiate tan x − cot x.
5. Differentiate e^5x.
6. Differentiate ln(3x).
7. Differentiate y = x³ sin x using the product rule.
8. Differentiate y = x/(x²+1) using the quotient rule.
9. Differentiate y = (2x+3)⁶ using the chain rule.
10. Differentiate y = sin(4x²).
11. Differentiate y = tan⁻¹(3x).
12. If y = x⁴ − 2x³ + x, find dy/dx at x = 1.

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21. PDF Notes

You can use the PDF below for quick revision of Differentiation and Basic Derivatives.

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22. Frequently Asked Questions – Differentiation

Q1. What is differentiation?

Differentiation is the mathematical process of finding the rate at which a function changes with respect to its variable.

Q2. What is the derivative of a constant?

The derivative of a constant is zero:

d/dx(c) = 0

Q3. What is the power rule?

The power rule states:

d/dx(xⁿ) = nxⁿ⁻¹

Q4. When is the product rule used?

Use the product rule when the function is expressed as the product of two differentiable functions:

(uv)′ = u′v + uv′

Q5. When is the quotient rule used?

Use the quotient rule when one function is divided by another:

(u/v)′ = (vu′ − uv′)/v²

Q6. When is the chain rule used?

The chain rule is used when one function is inside another function, such as sin(x²) or (3x²+1)⁵.

The basic shortcut is:

Outer derivative × Inner derivative

Q7. What is the derivative of sin x?

It is:

cos x

Q8. What is the derivative of cos x?

It is:

−sin x

Q9. What is the derivative of e^x?

The derivative of e^x is:

e^x

Q10. What is the derivative of ln x?

The derivative is:

1/x

Q11. What is the derivative of sin⁻¹x?

It is:

1/√(1−x²)

Q12. What is the most important habit while solving basic differentiation questions?

First identify the structure of the function and then choose the correct rule. Do not apply product, quotient or chain rules unnecessarily.

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23. Final Thoughts

Differentiation is one of the most important foundations of Calculus. The objective at the basic level is to become so familiar with the standard derivatives and rules that you can identify the correct method almost instantly.

The most important formulas to master are:

d/dx(xⁿ) = nxⁿ⁻¹

d/dx(sin x) = cos x

d/dx(cos x) = −sin x

d/dx(e^x) = e^x

d/dx(ln x) = 1/x

Along with these standard derivatives, the product rule, quotient rule and chain rule are essential tools for more complicated functions.

The best way to prepare is not just to memorize the formulas but to practice identifying which rule is required in each expression.

🔥 Golden Rule: Basic Function → Identify Rule → Differentiate → Simplify.

Once these basics become automatic, you will have a strong foundation for the more advanced differentiation problems that follow in JEE Mathematics.