IIT JEE Main Maths -Unit 7- Derivatives of Function- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 7- Derivatives of Function – Study Notes – New syllabus
IIT JEE Main Maths -Unit 7- Derivatives of Function – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Derivatives of Standard Functions
Differentiation: Rules (Sum, Product, Quotient, Chain)
Differentiation is the process of finding the derivative (rate of change) of a function. Once the standard derivatives are known, more complex functions can be differentiated using a set of algebraic rules.
Derivative of a Constant
$ \dfrac{d}{dx}(c) = 0 $
Example: \( \dfrac{d}{dx}(7) = 0 \)
Derivative of a Constant Multiple
If \( f(x) \) is differentiable and \( k \) is constant, then:
$ \dfrac{d}{dx}[k f(x)] = k \dfrac{d}{dx}[f(x)] $
Example: \( \dfrac{d}{dx}(5x^3) = 5(3x^2) = 15x^2 \)
Sum and Difference Rule
If \( f(x) \) and \( g(x) \) are differentiable, then:
$ \dfrac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $ $ \dfrac{d}{dx}[f(x) – g(x)] = f'(x) – g'(x) $
Example: \( \dfrac{d}{dx}(x^2 + \sin x) = 2x + \cos x \)
Product Rule
If \( f(x) \) and \( g(x) \) are differentiable, then:
$ \dfrac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) $
Example: \( \dfrac{d}{dx}(x^2 \sin x) = 2x\sin x + x^2\cos x \)
Quotient Rule
If \( f(x) \) and \( g(x) \) are differentiable and \( g(x) \ne 0 \), then:
$ \dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2} $
Example: \( \dfrac{d}{dx}\left(\dfrac{x^2}{\sin x}\right) = \dfrac{2x\sin x – x^2\cos x}{\sin^2 x} \)
Chain Rule (Derivative of Composite Functions)
If \( y = f(g(x)) \), where both \( f \) and \( g \) are differentiable, then:
$ \dfrac{dy}{dx} = f'(g(x)) \cdot g'(x) $
In words: Differentiate the outer function, then multiply by the derivative of the inner function.
Example: If \( y = \sin(x^2) \), then $ \dfrac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2) $
Extended Chain Rule
If \( y = f(g(h(x))) \), then:
$ \dfrac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) $
Logarithmic Differentiation (Special Case)
When both the base and the power are functions of \( x \), e.g., \( y = [f(x)]^{g(x)} \):
Take natural logs on both sides:
$ \ln y = g(x) \ln f(x) $
Then differentiate using product and chain rules.
| Differentiation Rules | |
| Constant Rule | \( \dfrac{d}{dx}[c] = 0 \) |
| Power Rule | \( \dfrac{d}{dx} x^n = nx^{n-1} \) |
| Product Rule | \( \dfrac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \) |
| Quotient Rule | \( \dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{g(x)f'(x) – f(x)g'(x)}{[g(x)]^2} \) |
| Chain Rule | \( \dfrac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \) |
Example
Find \( \dfrac{d}{dx}(x^2 + 3x^3) \).
▶️ Answer / Explanation
Using sum rule: differentiate each term.
\( \dfrac{d}{dx}(x^2) = 2x \), \( \dfrac{d}{dx}(3x^3) = 9x^2 \).
Answer: \( 2x + 9x^2 \)
Example
Find \( \dfrac{d}{dx}\left(\dfrac{x^2 + 1}{x^2 – 1}\right) \).
▶️ Answer / Explanation
Let \( f(x) = x^2 + 1 \), \( g(x) = x^2 – 1 \).
\( f'(x) = 2x, \, g'(x) = 2x \).
Using quotient rule: \( \dfrac{f'(x)g(x) – f(x)g'(x)}{(g(x))^2} \)
= \( \dfrac{2x(x^2 – 1) – (x^2 + 1)(2x)}{(x^2 – 1)^2} = \dfrac{-4x}{(x^2 – 1)^2} \).
Answer: \( \dfrac{-4x}{(x^2 – 1)^2} \)
Example
Find \( \dfrac{d}{dx}(\sin^2(3x^2 + 1)) \).
▶️ Answer / Explanation
Let \( y = [\sin(3x^2 + 1)]^2 \).
Apply chain rule: \( \dfrac{dy}{dx} = 2\sin(3x^2 + 1)\cos(3x^2 + 1) \cdot 6x \).
Simplify using identity \( 2\sin\theta\cos\theta = \sin(2\theta) \):
\( \dfrac{dy}{dx} = 6x \sin(2(3x^2 + 1)) = 6x\sin(6x^2 + 2) \).
Answer: \( 6x\sin(6x^2 + 2) \)
Notes and Study Materials
- Concepts of Differentiation
- Differentiation Master File
- Differentiation Revision Notes
- Differentiation Formulae
- Differentiation Reference Book
- Differentiation Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Differentiation” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Differentiation of the sum, difference, product, and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order up to two.
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