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
Derivatives of Standard Functions
Polynomial Functions
If \( f(x) = x^n \), where \( n \) is any real number, then:
$ \dfrac{d}{dx}(x^n) = n x^{n – 1} $
Example
Find \( \dfrac{d}{dx}(x^5) \).
▶️ Answer / Explanation
\( \dfrac{d}{dx}(x^5) = 5x^4 \).
Answer: \( 5x^4 \)
Example
Find \( \dfrac{d}{dx}(3x^4 – 5x^2 + 2x – 7) \).
▶️ Answer / Explanation
Apply power rule term by term:
\( = 3(4x^3) – 5(2x) + 2 – 0 = 12x^3 – 10x + 2 \).
Answer: \( 12x^3 – 10x + 2 \)
Trigonometric Functions
Standard derivatives:
| Function | Derivative |
|---|---|
| \( \sin x \) | \( \cos x \) |
| \( \cos x \) | \( -\sin x \) |
| \( \tan x \) | \( \sec^2 x \) |
| \( \cot x \) | \( -\csc^2 x \) |
| \( \sec x \) | \( \sec x \tan x \) |
| \( \csc x \) | \( -\csc x \cot x \) |
Example
Find \( \dfrac{d}{dx}(\sin x + \cos x) \).
▶️ Answer / Explanation
\( \dfrac{d}{dx}(\sin x) = \cos x, \; \dfrac{d}{dx}(\cos x) = -\sin x \).
Answer: \( \cos x – \sin x \)
Example
Find \( \dfrac{d}{dx}(\tan x \cdot \sec x) \).
▶️ Answer / Explanation
Use product rule:
\( (\tan x \sec x)’ = (\tan x)’ \sec x + \tan x (\sec x)’ \)
\( = \sec^2 x \sec x + \tan x (\sec x \tan x) = \sec^3 x + \sec x \tan^2 x \)
Answer: \( \sec^3 x + \sec x \tan^2 x \)
Exponential Functions
Standard results:
- \( \dfrac{d}{dx}(e^x) = e^x \)
- \( \dfrac{d}{dx}(a^x) = a^x \ln a \)
Example
Find \( \dfrac{d}{dx}(e^x + 3^x) \).
▶️ Answer / Explanation
\( \dfrac{d}{dx}(e^x) = e^x \), \( \dfrac{d}{dx}(3^x) = 3^x \ln 3 \).
Answer: \( e^x + 3^x \ln 3 \)
Example
Find \( \dfrac{d}{dx}(e^{2x^2}) \).
▶️ Answer / Explanation
Use chain rule: \( f(x) = e^{g(x)} \Rightarrow f'(x) = e^{g(x)} g'(x) \).
Here, \( g(x) = 2x^2 \Rightarrow g'(x) = 4x \).
Answer: \( 4x e^{2x^2} \)
Logarithmic Functions
Standard derivatives:
- \( \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \)
- \( \dfrac{d}{dx}(\log_a x) = \dfrac{1}{x \ln a} \)
Example
Find \( \dfrac{d}{dx}(\ln x) \).
▶️ Answer / Explanation
\( \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \).
Answer: \( \dfrac{1}{x} \)
Example
Find \( \dfrac{d}{dx}(\ln(3x^2 + 2)) \).
▶️ Answer / Explanation
Use chain rule: \( \dfrac{1}{3x^2 + 2} \cdot \dfrac{d}{dx}(3x^2 + 2) = \dfrac{6x}{3x^2 + 2} \).
Answer: \( \dfrac{6x}{3x^2 + 2} \)
Inverse Trigonometric Functions
Standard results:
| Function | Derivative |
|---|---|
| \( \sin^{-1} x \) | \( \dfrac{1}{\sqrt{1 – x^2}} \) |
| \( \cos^{-1} x \) | \( -\dfrac{1}{\sqrt{1 – x^2}} \) |
| \( \tan^{-1} x \) | \( \dfrac{1}{1 + x^2} \) |
| \( \cot^{-1} x \) | \( -\dfrac{1}{1 + x^2} \) |
| \( \sec^{-1} x \) | \( \dfrac{1}{|x|\sqrt{x^2 – 1}} \) |
| \( \csc^{-1} x \) | \( -\dfrac{1}{|x|\sqrt{x^2 – 1}} \) |
Example
Find \( \dfrac{d}{dx}(\sin^{-1} x) \).
▶️ Answer / Explanation
\( \dfrac{d}{dx}(\sin^{-1} x) = \dfrac{1}{\sqrt{1 – x^2}} \).
Answer: \( \dfrac{1}{\sqrt{1 – x^2}} \)
Example
Find \( \dfrac{d}{dx}(\tan^{-1}(3x)) \).
▶️ Answer / Explanation
Using chain rule: \( \dfrac{1}{1 + (3x)^2} \cdot 3 = \dfrac{3}{1 + 9x^2} \).
Answer: \( \dfrac{3}{1 + 9x^2} \)
Quick Summary Table (for Revision)
| Category | Key Formula |
|---|---|
| Polynomial | \( \dfrac{d}{dx}(x^n) = n x^{n – 1} \) |
| Trigonometric | \( \dfrac{d}{dx}(\sin x) = \cos x, \; \dfrac{d}{dx}(\cos x) = -\sin x, \; \dots \) |
| Exponential | \( \dfrac{d}{dx}(e^x) = e^x, \; \dfrac{d}{dx}(a^x) = a^x \ln a \) |
| Logarithmic | \( \dfrac{d}{dx}(\ln x) = \dfrac{1}{x} \) |
| Inverse Trig | \( \dfrac{d}{dx}(\sin^{-1} x) = \dfrac{1}{\sqrt{1 – x^2}}, \dots \) |