Calculus Formulas | Differentiation | Integration

Calculus is a branch of mathematics that deals with the study of rates of change (differential calculus) and the accumulation of quantities (integral calculus). It is divided into two main parts:

Differential Calculus: Focuses on the concept of the derivative, which allows us to determine the rate at which quantities change.
Integral Calculus: Involves the integral, which represents the accumulation of quantities and is used to compute areas, volumes, and other concepts that arise from adding up infinitesimal data points.

Basic Differentiation Formulas

Differentiation is the process of finding the derivative of a function, which represents its rate of change. Below is the list of basic differentiation formulas along with their definitions.

Differentiation Formulas
\frac{d}{dx}c = 0, where "c" is a constant.
\frac{d}{dx}x = 1
\frac{d(cx)}{dx} = c, where "c" is a constant and x is a variable.
\frac{d(x^n)}{dx} = nx^{n-1} , n ≠ 0.

Exponential and Logarithmic Derivatives

The differentiation of exponential and logarithmic functions focuses on their unique properties in calculus, specifically how they change with respect to their base variables. Here are the essential formulas for finding the derivatives of these functions, which are crucial for many applications in science and engineering:

Exponential and Logarithmic Derivatives
\frac{d}{dx}e^x = e^x
\frac{d}{dx}a^x = a^x \ln a, where a > 0
\frac{d}{dx}\ln x = \frac{1}{x} , where x > 0
\frac{d}{dx}log_a \ x \ = \frac{1}{x \ln a} , where x > 0, a> 0, a ≠ 1

Trigonometric Derivatives

Differentiation of trigonometric functions are listed below:

Trigonometric Functions
\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 \cdot \tan x
\frac{d}{dx} \csc x = -\csc x \cdot \cot x

Inverse Trigonometric Derivatives

Differentiation of inverse trigonometric functions is listed below:

Inverse Trigonometric Functions
\frac{d}{dx} [\sin^{-1} x] = \frac{1}{\sqrt{1 - x^2}} , x ≠ ±1
\frac{d}{dx} [\cos^{-1} x] = \frac{-1}{\sqrt{1 - x^2}} , x ≠ ±1
\frac{d}{dx}[ \tan^{-1} x] = \frac{1}{{1 + x^2}}
\frac{d}{dx}[ \cot^{-1} x] = \frac{-1}{1 + x^2}
\frac{d}{dx} [\sec^{-1} x] = \frac{1}{\|x\|\sqrt{x^2-1}} , x ≠ ±1, 0
\frac{d}{dx} [\csc^{-1} x] = \frac{-1}{\|x\|\sqrt{x^2-1}} , x ≠ ±1, 0

Read More about Derivative of Inverse Trigonometric Functions

Differentiation Rules

Differentiation rules for basic and composite functions are listed below:

Rule	Derivative Formula
Power Rule	\frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}
Constant Rule	\frac{d}{dx} \left( c \right) = 0
Constant Multiple Rule	\frac{d}{dx} \left( c \cdot g(x) \right) = c \cdot \frac{d}{dx} \left( g(x) \right)
Sum/Difference Rule	\frac{d}{dx} \left( g(x) \pm h(x) \right) = \frac{d}{dx} \left( g(x) \right) \pm \frac{d}{dx} \left( h(x) \right)
Product Rule	\frac{d}{dx} \left( g(x) \cdot h(x) \right) = \frac{d}{dx} \left( g(x) \right) \cdot h(x) + g(x) \cdot \frac{d}{dx} \left( h(x) \right)
Quotient Rule	\frac{d}{dx} \left( \frac{g(x)}{h(x)} \right) = \frac{ \frac{d}{dx} \left( g(x) \right) \cdot h(x) - g(x) \cdot \frac{d}{dx} \left( h(x) \right) }{h(x)^2}
Rule	\frac{d}{dx} \left( g(h(x)) \right) = \frac{d}{dx} \left( g(h(x)) \right) \cdot \frac{d}{dx} \left( h(x) \right)

Hyperbolic Functions

Derivatives of hyperbolic functions are listed below:

Derivatives of Hyperbolic Functions
\frac{d}{dx}[\sinh (x)] = \cosh(x)
\frac{d}{dx} \left[ \cosh(x) \right] = \sinh(x)
\frac{d}{dx} \left[ \tanh(x) \right] = \text{sech}^2(x)
\frac{d}{dx} \left[ \coth(x) \right] = -\text{csch}^2(x)
\frac{d}{dx} \left[ \text{sech}(x) \right] = -\text{sech}(x) \cdot \tanh(x)
\frac{d}{dx} \left[ \text{csch}(x) \right] = -\text{csch}(x) \cdot \cot(x)

Also Read:

Basic Integration Formulas

Integration is the process of finding the integral of a function, which represents the accumulation of quantities over a certain interval. Below is the list of basic integration formulas along with their definitions.

Property	Integration Formulas
Constant	∫ c dx = c · x + C
Power of x (for n ≠ -1)	\int x^n \ dx = \frac{x^{n+1}}{{n+1}} + \text{C}
Exponential Function	∫ e^x dx = e^x + C
Exponential Function with a constant base (a > 0, a ≠ 1)	\int a^x dx = \frac{a^x}{\ln (a)} + C

Common Integrals

Common integration formulas for polynomials are listed below:

Polynomials Formulas and Rational Function Integration Formulas
∫ dx = x + c
∫ k dx = kx + c
\int x^n \ dx = \frac{1}{n+1}x^{n+1}+c, n \neq -1
∫\frac{1}{x}dx = \ln \|x\| + c
∫ x⁻¹ dx = ln \|x\| + c
\int x^{-n} \ dx = \frac{1}{-n+1}x^{-n+1} + c, n \neq 1
\int \frac{1}{ax+b} \ dx = \frac{1}{a}\ln\|ax+b\|+c
\int x^{\frac{p}{q}} dx = \frac{1}{\frac{p}{q}+1}x^{\frac{p}{q}+1} + c =\frac{q}{p+q}x^{\frac{p+q}{q}}+c

Integration of Trigonometric Functions

Integration formulas for trigonometric functions are listed below:

Trigonometry integration Formulas
∫ cos(x) dx = sin(x) + c
∫ sin(x) dx = − cos(x) + c
∫ sec² x dx = tan(x) + c
∫ sec(x) tan(x) dx = sec(x) + c
∫ csc(x) cot(x) dx = − csc(x) + c
∫ csc² x dx = − cot(x) + c
∫ tan(x) dx = − ln cos(x) + c = ln sec(x) + c
∫ cot(x) dx = ln sin(x) + c = − ln csc(x) + c
∫ sec(x) dx = ln sec(x) + tan(x) + c
∫ sec³ (x) dx = ½( sec(x) tan(x) + ½ ln\| sec(x) + tan(x) \| ) + c
∫ csc(x) dx = ln csc(x) − cot(x) + c
∫ csc³ (x) dx = ½( −csc(x) cot(x) + ln \| csc(x) − cot(x) \|) + c

Read More about: Integration of Trigonometric Functions.

Inverse Trigonometric Integras

Integration formulas involving inverse trigonometric functions are listed below:

Inverse Trigonometric Integrals
∫sin ⁻¹x dx = x sin⁻¹x + (√1 − x²) + C
∫cos ⁻¹x dx = x cos⁻¹x - (√1 − x²) + C
∫tan ⁻¹x dx = x tan⁻¹x - ½ ln \|1 + x²\| + C
∫csc ⁻¹x dx = x csc⁻¹x + ln \|x + (√x² - 1)\| + C
∫sec ⁻¹x dx = x sec⁻¹x - ln \|x + (√x² - 1)\| + C
∫cot ⁻¹x dx = x cot⁻¹x + ½ ln \|1 + x²)\| + C

Exponential And Logarithmic Functions

Integration formulas for exponential and logarithmic functions are listed below:

Integration Formulas for Exponential & Logarithmic Functions
∫ e^xdx = e^x+ c
∫ a^xdx = ax ln(a) + c
∫ ln (x) dx =x ln (x) − x + c
\int e^{ax} \sin(bx) \, dx = e^{ax} \left( \frac{a \sin(bx) - b \cos(bx)}{a^2 + b^2} \right) + c
∫ x e^x dx = (x − 1)e^x+ c
\int e^{ax} \cos(bx) \, dx = e^{ax} \left( \frac{a \cos(bx) + b \sin(bx)}{a^2 + b^2} \right) + c
\int \frac{1}{x} \ln(x) \, dx = \ln(x) \cdot \ln(x) - 2 \int \ln(x) \, dx + c

Integration of Special Functions

Special integrals involving unique functions are listed below:

Special Integrals
\quad \int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left\| \frac{x-a}{x+a} \right\| + C
\quad \int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log \left\| \frac{a+x}{a-x} \right\| + C
\quad \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \frac{x}{a} + C
\quad \int \frac{dx}{\sqrt{x^2 - a^2}} = \log \left\| x + \sqrt{x^2 - a^2} \right\| + C
\quad \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \frac{x}{a} + C
\quad \int \frac{dx}{\sqrt{x^2 + a^2}} = \log \left\| x + \sqrt{x^2 + a^2} \right\| + C

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and is given by:

∫ f(x) g'(x) dx = f(x) g(x) – ∫ g(x) f'(x)dx

For functions u and v, it can also be written as:

∫u dv = uv − ∫v du

Where u and dv are differentiable functions of x.

Limits and Continuity

It provides formulas of limits and continuity, which are the backbone of understanding how functions behave near specific points.

Basic Limits: Some of the the basic formulas related to limits are:

lim_{x ⇢ a} k = k, where k is a constant quantity
lim_{x ⇢ a} x = a
lim_{x ⇢ a} bx + c = ba + c
lim _{x ⇢ a} xⁿ = aⁿ if n is a positive integer.
lim _{x ⇢ +0} 1/x^r = +∞
lim _{x ⇢ −0} 1/x^r = −∞, if r is odd
lim _{x ⇢ −0} 1/x^r = +∞, if r is even
lim_x⇢a (xⁿ – aⁿ)/(x – a) = na^(n-1)
lim_x⇢a sin x/x = 1
lim_x⇢a tan x/x = 1
lim_x⇢a(1 – cos x)/x = 0
lim_x⇢a cos x = 1
lim_x⇢a e^x = 1
lim_x⇢a (e^x – 1)/x = 1
lim_x⇢a (1 + 1/x)^x = e

L'Hôpital's Rule: L'Hospital's Rule is a method for finding limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of a quotient of two functions results in an indeterminate form, the limit of their derivatives can be used to find the original limit.

Suppose lim_⁡x→cf(x) = 0 and lim⁡_x→cg(x) = 0 or lim_⁡x→cf(x) = ±∞ and lim⁡_x→cg(x) = ±∞.

Then, if the necessary condition hold,

lim _{x → a} f(x)/g(x) = lim _{x → a} f'(x)/g'(x) = lim _{x → a} f”(x)/g”(x)= . . .
provided that the right-hand limits exist or are infinite.

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Practical Applications of Calculus

Engineering: Calculus is used to design components and analyze rates of change in civil, mechanical, and electrical engineering, such as calculating stress on materials and optimizing electrical flows.
Physics: Essential for modeling dynamics, deriving equations of motion, solving electromagnetic equations, and analyzing quantum mechanics.
Economics: Helps in modeling economic growth, optimizing cost and output functions, and calculating demand elasticity to predict economic trends.
Medicine: Utilized for modeling disease spread, optimizing drug dosages with variable-rate infusion pumps, and understanding changes in human physiology.
Computer Science: Fundamental in optimizing algorithms in machine learning, rendering simulations in graphics, and modeling trends in data science.

Read in detail: Applications of Calculus.

Calculus Formulas | Differentiation | Integration | Limits