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Differentiation Formulas

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Team eSaral

eSaral Academic and Editorial Team

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Summary

The differentiation formula is the standard rule used to find the derivative of a function — for example, $\frac{d}{dx}(x^n)=nx^{n-1}$ and $\frac{d}{dx}(e^x)=e^x$. This differential formula list covers algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions, along with the product, quotient, and chain rules, forming the base of the entire Continuity and Differentiability chapter in Class 12 Mathematics and JEE.

Differentiation Formula: Complete Formulas for Derivatives

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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: 

Used when the function has the form $y=[f(x)]^{g(x)}$, such as $y=x^{\sin x}$. Take the natural logarithm of both sides before differentiating, since standard differentiation rules alone cannot directly handle a variable exponent.

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

$\frac{d}{dx}\left(\frac{x}{\sin x}\right)=\frac{\sin x-x\cos x}{\sin^2 x}$

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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Frequently Asked Questions

What are the formulas differentiation for trigonometric functions?

The formulas differentiation for trigonometric functions include $\frac{d}{dx}(\sin x)=\cos x$, $\frac{d}{dx}(\cos x)=-\sin x$, and $\frac{d}{dx}(\tan x)=\sec^2 x$, among others.

What is the differential formula for $e^x$?

The differential formula for $e^x$ is $\frac{d}{dx}(e^x)=e^x$, since $e^x$ is the unique function whose derivative equals itself.

What is the formula for differentiation of $x^n$?

The formula for differentiation of $x^n$ is $\frac{d}{dx}(x^n)=nx^{n-1}$, known as the power rule, and it is the most commonly used differentiation formula in calculus.

What is the differentiation formula?

The differentiation formula is a standard rule used to find the derivative of a function, such as $\frac{d}{dx}(x^n)=nx^{n-1}$, representing the rate of change of that function.

How many differentiation formulas should I memorise for Class 12 boards?

For Class 12 boards, roughly 15–20 standard differentiation formulas are sufficient, covering algebraic, exponential, logarithmic, trigonometric, and inverse trigonometric functions, along with the product, quotient, and chain rules.

When is logarithmic differentiation used instead of the standard differentiation formula?

Logarithmic differentiation is used when a function has a variable base raised to a variable exponent, such as $y=x^{\sin x}$, since the standard differentiation formula list cannot be applied to such forms directly.

What is the chain rule differentiation formula?

The chain rule differentiate formula is $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$, used when differentiating a composite function, such as $\sin(3x^2)$.

What is the product rule in the differentiation formula list?

The product rule states $\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$, used to differentiate the product of two functions $u$ and $v$.

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eSaral Academic and Editorial Team

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