Differentiation Formulas
Table of Contents
- Differentiation Formula List – Standard Derivative Formulas
- Basic Algebraic Differentiation Formulas
- Exponential and Logarithmic Differentiation Formulas
- What Is the Differentiation Formula List for Trigonometric Functions?
- What Are the Inverse Trigonometric Differentiation Formulas?
- What Are the Main Rules of Differentiation?
- What Is Logarithmic and Implicit Differentiation?
- Solved Examples Using the Differentiation Formula
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Differentiability / Derivatives chapter covers the complete differentiation formula list — algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic — along with the key rules of differentiation and solved examples.
Differentiation Formula List – Standard Derivative Formulas
Every differentiation formula in this list represents the rate of change of a function with respect to its variable. This differentiation formula list is the foundation of every topic that follows — Application of Derivatives, Integrals, and Differential Equations all depend on fluency here.
Basic Algebraic Differentiation Formulas
| Function | Differentiation Formula |
|---|---|
| $x^n$ | $\frac{d}{dx}(x^n)=nx^{n-1}$ |
| $x$ | $\frac{d}{dx}(x)=1$ |
| $c$ (constant) | $\frac{d}{dx}(c)=0$ |
| $cx$ (constant multiple) | $\frac{d}{dx}(cf(x))=c\frac{d}{dx}f(x)$ |
| $\sqrt{x}$ | $\frac{d}{dx}(\sqrt{x})=\frac{1}{2\sqrt{x}}$ |
| $\frac{1}{x}$ | $\frac{d}{dx}\left(\frac{1}{x}\right)=-\frac{1}{x^2}$ |
Exponential and Logarithmic Differentiation Formulas
| Function | Differentiation Formula |
| $e^x$ | $\frac{d}{dx}(e^x)=e^x$ |
| $a^x$ | $\frac{d}{dx}(a^x)=a^x\ln a$ |
| $\ln x$ | $\frac{d}{dx}(\ln x)=\frac{1}{x}$ |
| $\log_a x$ | $\frac{d}{dx}(\log_a x)=\frac{1}{x\ln a}$ |
What Is the Differentiation Formula List for Trigonometric Functions?
The differentiation formula for trigonometric functions is one of the most frequently tested parts of this differentiation formula list, since these results recur across almost every JEE calculus question.
Trigonometric Differentiation Formulas
| Function | Differentiation Formula |
| $\sin x$ | $\frac{d}{dx}(\sin x)=\cos x$ |
| $\cos x$ | $\frac{d}{dx}(\cos x)=-\sin x$ |
| $\tan x$ | $\frac{d}{dx}(\tan x)=\sec^2 x$ |
| $\cot x$ | $\frac{d}{dx}(\cot x)=-\csc^2 x$ |
| $\sec x$ | $\frac{d}{dx}(\sec x)=\sec x\tan x$ |
| $\csc x$ | $\frac{d}{dx}(\csc x)=-\csc x\cot x$ |
Test your recall speed with JEE Main Chapterwise PYQ, where trigonometric differentiation appears in nearly every paper.
What Are the Inverse Trigonometric Differentiation Formulas?
Inverse trigonometric derivative formulas complete this differentiate formula list and appear regularly in composite-function and implicit differentiation problems.
Inverse Trigonometric Differentiation Formulas
| Function | Differentiation Formula |
| $\sin^{-1}x$ | $\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}$ |
| $\cos^{-1}x$ | $\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^2}}$ |
| $\tan^{-1}x$ | $\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}$ |
| $\cot^{-1}x$ | $\frac{d}{dx}(\cot^{-1}x)=-\frac{1}{1+x^2}$ |
| $\sec^{-1}x$ | $\frac{d}{dx}(\sec^{-1}x)=\frac{1}{x\sqrt{x^2-1}}$ |
| $\csc^{-1}x$ | $\frac{d}{dx}(\csc^{-1}x)=-\frac{1}{x\sqrt{x^2-1}}$ |
Practice these directly with JEE Advanced Chapterwise PYQ, where inverse trig derivatives often show up inside harder composite functions.
What Are the Main Rules of Differentiation?
Beyond the standard differentiation formula for individual functions, three core rules of differentiation are used to differentiate combinations of functions:
Product, Quotient, and Chain Rule
| Rule | Formula for Differentiation |
| Product Rule | $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$ |
| Quotient Rule | $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$ |
| Chain Rule | $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$ |
These three rules — especially the Chain Rule — form the backbone of nearly every calculus question in JEE Main PYQ, so fluency here directly impacts speed on the rest of the paper.
What Is Logarithmic and Implicit Differentiation?
Some functions cannot be differentiated directly using the standard differentiation formula list, and require an additional step first:
Logarithmic differentiation:
Implicit differentiation:
Used when $y$ is not directly expressed as a function of $x$, such as in $x^2+y^2=r^2$. Differentiate both sides with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule wherever $y$ appears.
Once comfortable with both methods, test your application under timed conditions using eSaral's JEE Test Series, which includes dedicated Calculus mock sections.
Solved Examples Using the Differentiation Formula
Applying the differentiation formula, once the pattern is recognised, follows direct substitution into the matching standard result:
Example 1: Differentiate $f(x)=x^5$
$\frac{d}{dx}(x^5)=5x^4$
Example 2: Differentiate $f(x)=x^2\sin x$ using the product rule
$\frac{d}{dx}(x^2\sin x)=x^2\cos x+2x\sin x$
Example 3: Differentiate $f(x)=\sin(3x^2)$ using the chain rule
Let $u=3x^2$, so $\frac{du}{dx}=6x$.
$\frac{d}{dx}(\sin(3x^2))=\cos(3x^2)\times6x=6x\cos(3x^2)$
Example 4: Differentiate $f(x)=\frac{x}{\sin x}$ using the quotient rule
$\frac{d}{dx}\left(\frac{x}{\sin x}\right)=\frac{\sin x-x\cos x}{\sin^2 x}$
Explore More on eSaral (Related Continuity, Differentiability & Class 12 Mathematics Resources)
Master the complete chapter this topic belongs to: Continuity and Differentiation Notes for Class 12 & IIT JEE →
Apply these formulas to real problems: Application of Derivatives Class 12 Notes →
See where derivatives are used next: Differential Equations Class 12 Notes →
Quick revision before your exam: Mind Maps for Methods of Differentiation →
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