Mathematics • Calculus • Formula Vault

All Differentiation & Integration Formulas

A comprehensive, exam-ready reference of the standard differentiation and integration formulas used in senior-secondary mathematics, CBSE, JEE and other competitive examinations.

How to use this page: Learn the basic formulas first, then the rules, standard functions, methods of integration and finally the definite-integral and reduction formulas. The page is designed as a single revision reference rather than a short list.

1. Basic Differentiation Formulas

Function	Derivative
Constant
x
xⁿ
√x
1/x
c f(x)
f(x) ± g(x)

2. Fundamental Rules of Differentiation

Rule	Formula
Product rule
Quotient rule
Chain rule
Composite power
General function of u
First-principles definition

3. Algebraic, Exponential and Logarithmic Derivatives

Function	Derivative
e^x
e^u
a^x, a > 0, a ≠ 1
a^u
ln x
ln\|u\|
log_ax
u^u
x^x

Logarithmic differentiation: For a positive function y, take logarithms first. For example, if y = [f(x)]^g(x), then and hence .

4. Trigonometric Differentiation Formulas

Function	Derivative
sin x
cos x
tan x
cot x
sec x
csc x
sin u
cos u
tan u
cot u
sec u
csc u

5. Inverse Trigonometric Derivatives

Function	Derivative
sin⁻¹x
cos⁻¹x
tan⁻¹x
cot⁻¹x
sec⁻¹x
csc⁻¹x

For a composite argument u, multiply each corresponding derivative by u'. For example:

\frac{d}{dx}\left[\sin^{-1}u\right]=\frac{u'}{\sqrt{1-u^2}}

6. Hyperbolic Function Derivatives

Function	Derivative
sinh x
cosh x
tanh x
coth x
sech x
csch x

7. Higher-Order, Parametric and Implicit Differentiation

Topic	Formula
Second derivative
nth derivative
Parametric first derivative
Parametric second derivative
Logarithmic differentiation
Leibniz rule

8. Standard Integration Rules

Function / expression	Integral
Constant
Power
1/x
f'(x)
Constant multiple
Sum/difference

9. Exponential and Logarithmic Integration

Expression	Integral
e^x
e^ax
a^x
a^ax+b
1/x
f'/f

10. Basic Trigonometric Integrals

Expression	Integral
sin x
cos x
sec²x
csc²x
sec x tan x
csc x cot x
tan x
cot x
sec x
csc x

11. Trigonometric Integrals with a Linear Argument

Integral	Result

12. Standard Integrals Leading to Inverse Trigonometric Functions

Integral	Result

13. Standard Quadratic-Denominator Integrals

Integral	Result / method

14. Integration by Substitution

If an integrand contains a function and its derivative, substitution is often the quickest method.

u=g(x),\quad du=g'(x)dx

\int f(g(x))g'(x)dx=\int f(u)du

Core pattern: Look for an “inside function” and its derivative. Replace the inside function by u, integrate with respect to u, and substitute back.

15. Integration by Parts

\boxed{\int u\,dv=uv-\int v\,du}

A useful choice rule is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. It is a heuristic for choosing u, not a mathematical theorem.

16. Integration of Important Products

Integral	Useful result

17. Important Integrals Involving sin²x and cos²x

Integral	Result

18. Reduction Formulas for Trigonometric Integrals

For positive integers n, the following standard reduction formulas are useful in advanced problems.

Integral	Reduction formula
I_n = ∫ sinⁿx dx
I_n = ∫ cosⁿx dx
I_n = ∫ secⁿx dx
I_n = ∫ cscⁿx dx
I_n = ∫ tanⁿx dx
I_n = ∫ cotⁿx dx

19. Standard Definite Integral Formula

\boxed{\int_a^b f(x)dx=F(b)-F(a),\quad F'(x)=f(x)}

This is the Fundamental Theorem of Calculus and connects antiderivatives with definite integrals.

Property	Formula
Same limits
Reversal of limits
Additivity
Linearity
Even function
Odd function
Symmetry substitution
Periodic function	when T is a period.

20. Definite Integrals with Symmetry

Useful result	Formula
Reflection property
Average of symmetric terms
Special 0 to π/2

21. Important Trigonometric Identities Used in Differentiation and Integration

Identity	Formula
Fundamental identity
Tangent identity
Cotangent identity
Double angle
Cosine double angle
Power reduction
Sum-to-product
Cosine sum-to-product

22. Hyperbolic Integration Formulas

Expression	Integral
sinh x
cosh x
sech²x
csch²x
sech x tanh x
csch x coth x

23. Common Derivative–Integral Pairs

A powerful revision technique is to learn formulas in pairs. If the derivative of F(x) is f(x), then F(x) is an antiderivative of f(x).

Derivative	Corresponding integral

24. Important Exam Notes

In trigonometric differentiation and integration, the standard calculus formulas use angles measured in radians.
For , the power formula requires n ≠ −1.
Always write +C for an indefinite integral.
For a definite integral, +C is not written after evaluating the limits.
For inverse-trigonometric derivative formulas, pay attention to domain restrictions and the absolute value in the sec⁻¹ and csc⁻¹ formulas.
For composite functions, do not forget the chain-rule factor u'.
For products, use the product rule; do not multiply the two derivatives.
For quotients, use the quotient rule or rewrite the expression as a product with a negative power when convenient.
Differentiate or integrate after simplifying the algebra whenever that makes the structure clearer.

25. Ultra-Quick Revision Sheet

Power:

Product:

Quotient:

Chain:

Exponential:

Logarithm:

Trigonometric:

Inverse trig:

Power integral:

Exponential integral:

1/x:

By parts:

Definite integral:

26. Scope of This Formula Vault

This page is intentionally much broader than a basic school formula list. It combines the standard single-variable calculus formulas most commonly required through senior-secondary mathematics and competitive-exam preparation: elementary derivatives, rules, trigonometric and inverse-trigonometric functions, logarithmic and exponential functions, hyperbolic functions, higher derivatives, parametric forms, standard antiderivatives, inverse-trigonometric integrals, integration techniques, reduction formulas and definite-integral properties.

Important: No finite page can literally contain every formula that can ever be derived in calculus; new formulas can be generated from these rules and identities. The purpose here is to provide a comprehensive standard reference rather than claim that every possible derived identity is separately listed.

CALCULUS FORMULA VAULT

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Use this page as a quick reference while solving differentiation and integration problems on Saraswat Academy.

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