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A comprehensive list of basic differentiation formulae and rules

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A comprehensive list of basic differentiation formulae and rules

A handy quick-reference list of all basic differentiation formulae and rules. Use these in your exam, school or while learning on Physics Capsule.

Differentiation is often how many students are introduced to calculus. Its lengthy list of formulae and methods and rules are almost intimidating and it helps to have them collectively in one place for reference.

This list is aimed at serving that purpose: it is not a guide or explanation, but a collection of formulae and rules structured to be easy to follow and to serve as a reference sheet to look through and recollect.

It is divided into two parts: identities, or the basic derivative formulae; and rules which explain the methods of solving common differentiation forms. We refer to variables as x, y, z etc. and functions performed on those variables as f(x), g(x) etc. with f \prime (x), g\prime (x) etc. being their respective derivatives. We will refer to constants as C, a or b and the exponent as e.


Differentiation formulae

The following are various formulae used to compute the derivative (i.e. to differentiate) a function. These can be derived from first principles, although we will not be doing that here.

Reading this table is simple: for instance, at (7) we have the following:

7

\tan x

\sec^2 x

This is simply ready as \frac{d}{dx}\tan x = \sec^2 x.

Basic

Sl. no.

1

2-a

2-b

3-a

3-b

4-a

4-b

5

f(x)

C

x^n

x^{-1} \text{ or } \frac{1}{x}

a^x, a>0

e^x

\ln x

\log_a x

x^x

f'(x)

0

n x^{n-1}

\frac{-1}{x^2}

a^x \ln a

e^x

\frac{1}{x}

\frac{1}{x \ln a}

x^x (1 + \ln x)

Trigonometric

6

7

8

9

10

11

\sin x

\cos x

\tan x

\sec x

\csc x

\cot x

\cos x

- \sin x

\sec^2 x

\sec x \tan x

- \csc x \cot x

- \csc^2 x

Inverse trigonometric

12

13

14

15

16

17

\sin^{-1} x

\cos^{-1} x

\tan^{-1} x

\sec^{-1} x

\csc^{-1} x

\cot^{-1} x

\frac{1}{\sqrt{1-x^2}}

\frac{-1}{\sqrt{1-x^2}}

\frac{1}{x^2 + 1}

\frac{1}{\left|x\right|\sqrt{x^2-1}}

\frac{-1}{\left|x\right|\sqrt{x^2-1}}

\frac{-1}{x^2 + 1}

Hyperbolic trigonometric

18

19

20

21

22

23

\sinh x

\cosh x

\tanh x

{\mathop{\rm sech}\nolimits} x

{\mathop{\rm csch}\nolimits} x

\coth x

\cosh x

\sinh x

{\mathop{\rm sech}\nolimits}^2 x

- \tanh x {\mathop{\rm sech}\nolimits} x

- \coth x {\mathop{\rm csch}\nolimits} x

- {\mathop{\rm csch}\nolimits}^2 x

Inverse hyperbolic trigonometric

24

25

26

27

28

29

\sinh^{-1} x

\cosh^{-1} x

\tanh^{-1} x

{\mathop{\rm sech}\nolimits}^{-1} x

{\mathop{\rm csch}\nolimits}^{-1} x

\coth^{-1} x

\frac{1}{\sqrt{x^2 + 1}}

\frac{1}{\sqrt{x^2- 1}}

\frac{1}{1-x^2}

\frac{-1}{\left|x\right| \sqrt{1 - x^2}}

\frac{1}{\left|x\right| \sqrt{1 + x^2}}

\frac{-1}{1-x^2}

Differentiation rules

There are some rules when differentiating more complicated functions or functions within functions. The idea is simply to compare the function(s) at hand with the rules, see which rule the function(s) best resembles and apply that rule. To solve specific functions, we once again have to look to the list of differentiation formulae above.

Rule

Adding/subtracting

 

Multiply constant

 

Product rule

 

Quotient rule

 

Logarithmic rule

 

Chain rule

Mechanism

\frac{d}{dx}\left[ f(x) \pm g(x) \right] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)

 

\frac{d}{dx} C f(x) = C \frac{d}{dx}f(x)

 

\frac{d}{dx}\left( f(x) \cdot g(x) \right) = f(x) \frac{d}{dx} g(x) + g(x) \frac{d}{dx} f(x)

 

\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f\prime (x) - f(x) g\prime (x)}{g(x)^2}

 

\frac{d}{dx} \ln f(x) = \frac{f \prime (x)}{f(x)}

 

\frac{d}{dx}f(g(x)) = f\prime (g(x)) \cdot g\prime (x) \; \text{or} \; \frac{d}{d(g(x))} f(g(x)) \frac{d}{dx}g(x)

These are the most commonly encountered rules and differentiation formulae in physics. If you look through our Learn section, you might find a lot of these formulae being used or referred to.

For convenience, a pdf version of this list will be made available for download soon.

Cover image by Alex.

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V.H. Belvadi is a physics postgraduate specialising in solid-state physics and also interested in astroparticle physics. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin.

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