Methods of Differentiation – Complete JEE Revision
Differentiation is about choosing the correct method, not brute force. JEE questions are designed to punish wrong method selection. This chapter systematically covers all differentiation methods used in JEE Mains & Advanced.
1. Basic Concept
\[
\frac{dy}{dx} = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}
\]
Differentiation measures the rate of change of one quantity with respect to another.
2. Standard Derivatives (Must Memorise)
- \(\frac{d}{dx}(x^n)=nx^{n-1}\)
- \(\frac{d}{dx}(e^x)=e^x\)
- \(\frac{d}{dx}(a^x)=a^x\ln a\)
- \(\frac{d}{dx}(\ln x)=\frac{1}{x}\)
- \(\frac{d}{dx}(\sin x)=\cos x\)
- \(\frac{d}{dx}(\cos x)=-\sin x\)
- \(\frac{d}{dx}(\tan x)=\sec^2 x\)
- \(\frac{d}{dx}(\cot x)=-\csc^2 x\)
- \(\frac{d}{dx}(\sec x)=\sec x\tan x\)
- \(\frac{d}{dx}(\csc x)=-\csc x\cot x\)
3. Method 1 – Differentiation from First Principle
Used mainly in conceptual or proof-based questions.
Example:
Differentiate \(f(x)=x^2\) \[ \frac{d}{dx}(x^2) = \lim_{h\to0}\frac{(x+h)^2-x^2}{h} =2x \]
Differentiate \(f(x)=x^2\) \[ \frac{d}{dx}(x^2) = \lim_{h\to0}\frac{(x+h)^2-x^2}{h} =2x \]
4. Method 2 – Chain Rule
If \(y=f(g(x))\), then:
\[
\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}
\]
Example:
Differentiate \(y=\sin(x^2)\) \[ \frac{dy}{dx}=\cos(x^2)\cdot2x \]
Differentiate \(y=\sin(x^2)\) \[ \frac{dy}{dx}=\cos(x^2)\cdot2x \]
5. Method 3 – Implicit Differentiation
Used when \(y\) is not explicitly written as a function of \(x\).
Example:
Given \(x^2+y^2=1\) \[ 2x+2y\frac{dy}{dx}=0 \Rightarrow \frac{dy}{dx}=-\frac{x}{y} \]
Given \(x^2+y^2=1\) \[ 2x+2y\frac{dy}{dx}=0 \Rightarrow \frac{dy}{dx}=-\frac{x}{y} \]
6. Method 4 – Logarithmic Differentiation
Used when variable appears in both base and power.
Example:
Differentiate \(y=x^x\) \[ \ln y=x\ln x \Rightarrow \frac{1}{y}\frac{dy}{dx}=\ln x+1 \] \[ \Rightarrow \frac{dy}{dx}=x^x(\ln x+1) \]
Differentiate \(y=x^x\) \[ \ln y=x\ln x \Rightarrow \frac{1}{y}\frac{dy}{dx}=\ln x+1 \] \[ \Rightarrow \frac{dy}{dx}=x^x(\ln x+1) \]
7. Method 5 – Parametric Differentiation
If \(x=f(t)\) and \(y=g(t)\):
\[
\frac{dy}{dx}=\frac{dy/dt}{dx/dt}
\]
Example:
\(x=t^2,\; y=t^3\) \[ \frac{dy}{dx}=\frac{3t^2}{2t}=\frac{3t}{2} \]
\(x=t^2,\; y=t^3\) \[ \frac{dy}{dx}=\frac{3t^2}{2t}=\frac{3t}{2} \]
8. Method 6 – Differentiation of Inverse Functions
\[
\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}
\]
\[
\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}
\]
9. Higher Order Derivatives
Repeated differentiation of a function.
Example:
If \(y=e^x\sin x\), \[ \frac{d^2y}{dx^2}=2e^x\cos x \]
If \(y=e^x\sin x\), \[ \frac{d^2y}{dx^2}=2e^x\cos x \]
10. JEE Traps & Pain Points
- Forgetting chain rule
- Wrong differentiation of inverse trig
- Ignoring domain restrictions
- Mixing implicit & explicit methods
- Sign mistakes in higher derivatives
Core Insight:
JEE does not test differentiation — it tests method selection. Recognize the form first, then apply the method.
JEE does not test differentiation — it tests method selection. Recognize the form first, then apply the method.
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