Calculus Formula: Limits, Differentials & Integrals

Calculus is a branch of mathematics that deals with the study of the “rate of change”. It is applied in solving equations. Calculus is also referred to as infinitesimal calculus or infinite calculus. The two major branches of calculus are differential calculus and integral calculus. Differential calculus is concerned with rates of change and slopes of curves, whereas Integral calculus is the branch dealing with the accumulation of quantities and the areas under and between curves. Using differential calculus, it is possible to determine velocity and acceleration in mechanics from the position function. 

Read More: Integration 

Key Terms: Calculus, Calculus Formula, Differential Calculus, Integral Calculus, Limits, Derivatives, Integration, Differentiation, Function, Integration

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What is Calculus Formula?

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Basic calculus consists of rules and differentiation formulas. This is the method that is used to calculate the derivative of a function. Integration is the method used to calculate the anti-derivative of a function. Calculus formulas describe the rate of change of a function for the given input value using the derivative of a function or differentiation formula. The process of finding the derivative of any given function is called differentiation.

Calculus

Read More: Double Integral

Integrals Detailed Video Explanation

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Limits Formula

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Limits of functions at a point are the common and coincidence value of the left and right-handed limits. It is denoted as:

where f(x) is a function of x.  

This formula means that,

If at a point x = a the function f(x) takes an indeterminate form, then the values of f(x) which is close to a are considered.  If the value tends to a definite value as x tends to a, then the value obtained is called the Minit of f(x) at x=a.

• The expected value of the function f(x) shown by the points to the left of point ‘a’ is the left-hand limit of the function at that point. It is denoted as:

• The point to the right of the point, that is, ‘a’, which generally shows the value of the function is the right-hand limit of the function at that point. It can be denoted as:

Note: The value of a limit of a function f(x) at a point, that is, f(a) may vary from the value of f(x) at the point ‘a’.

Given below is the list of formulae for calculating limits:

1. \(\lim\limits_{x \to 0}\)\(\frac{sinx}{x}\) = \(\lim\limits_{x \to 0}\)\(\frac{tanx}{x}\) = 1
2. \(\lim\limits_{x \to 0}\)\(\frac{sin^{-1}x}{x}\)= \(\lim\limits_{x \to 0}\)\(\frac{tan^{-1}x}{x}\)= 1
3. \(\lim\limits_{x \to 0}\)\(\frac{1n(1+x)}{x}\)= 1
4. \(\lim\limits_{x \to 0}\) \(\frac{a^x -1}{x}\)= 1na
5. \(\lim\limits_{x \to 0}\)\(\frac{e^x -1}{x}\)= 1
6. \(\lim\limits_{x \to 0}\)\(\frac{x^n - a^n}{(x-a)} = n.a^{n-1}\)
7. \(\lim\limits_{x \to 0}\)(1 + x)^{\(^{\frac{1}{x}}\)}= \(\lim\limits_{x \to 0}\)\((1 + \frac{1}{x})^x\)= c
8. \(\lim\limits_{x \to 0}\)\(\frac{(1+x)^m -1}{x} = m\)

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Properties of Limits

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The following figure summarises the properties of limits used for calculations:

If \(\lim\limits_{x \to a}\) f(x) = m and \(\lim\limits_{x \to a}\) g(x) = n

where ‘m’ and ’n’ are real and finite the

(i) \(\lim\limits_{x \to a}\) [f(x) = g(x)] = \(\lim\limits_{x \to a}\) f(x) = \(\lim\limits_{x \to a}\) g(x) = m ± n

(ii) \(\lim\limits_{x \to a}\) [cf(x)] = c . \(\lim\limits_{x \to a}\) f(x) = c.m

(iii) \(\lim\limits_{x \to a}\) (f(x) . g(x)) = \(\lim\limits_{x \to a}\) f(x) . \(\lim\limits_{x \to a}\) g(x) = m . n

(iv) \(\lim\limits_{x \to a}\)\(\frac{f(x)}{g(x)}\) = \(\frac{ \lim\limits_{x \to a} f(x)}{\lim\limits_{x \to a} g(x)}\) = \(\frac{m}{n}\), provided n ≠ 0

In the figure, f(x), and g(x) are functions, and ‘m’ and ‘n’ is real and finite constants. Properties of multiplication, division, addition or subtraction, and multiplication by a constant are provided above. 

Read More: Increasing & Decreasing Functions

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Differential Calculus Formula

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Differential calculus can be defined as the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. This involves the derivative equation that describes the rate at which a change occurs in a given function. Minimum or maximum values of given functions for optimization problems are one the prominent applications of differential calculus.

Given below is the list of differential calculus formulae that can be used to find the derivation of a function:

• \(\frac{d}{dx}\)xⁿ = nx^n-1
• \(\frac{d}{dx}\) (fg) = fg’ + gf’
• \(\frac{d}{dx}\)\((\frac{f}{g}) = \frac{gf'-fg'}{g^2}\)
• \(\frac{d}{dx}\)f(g(x)) = f’ (g(x))g’(x)
• \(\frac{d}{dx}\)(sin x) = cos x
• \(\frac{d}{dx}\)(cos x) = – sin x
• \(\frac{d}{dx}\)(tan x) = sec²x 
• \(\frac{d}{dx}\) (cot x) = – csc²x 
• \(\frac{d}{dx}\) (sec x) = sec x tan x
• \(\frac{d}{dx}\) (csc x) = – csc x cot x
• \(\frac{d}{dx}\) (e^x) = e^x
• \(\frac{d}{dx}\)(a^x) = a^x 1n a
• \(\frac{d}{dx}\) 1n x = \(\frac{1}{x}\)
• \(\frac{d}{dx}\) (arc sin x) = \(\frac{1}{\sqrt{1 - x^2}}\)
• \(\frac{d}{dx}\) (arc tan x) =\(\frac{1}{1 + x^2}\)

Read More: Multivariable Calculus

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Integral Calculus Formula

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Integral Calculus can be defined as the branch of calculus that is concerned with Integrals, the accumulation of quantities, and the areas under and between curves and their properties and their properties.

Given below is a list of Integral Calculus Formulae that can be used to find the integral of a function:

Read More: Properties of Definite Integral

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Differential Calculus vs Integral Calculus

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Differential calculus utilizes differentiation to find the derivative of the function whereas integral calculus employs integration to find the integral of the function. Integration is the reverse of differentiation. It is also known as anti-derivative. Some of the differences between differential calculus and integral calculus are tabulated below: 

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Things to Remember

• Calculus formulae describe the rate of change of a function for the given input value using the derivative of a function or differentiation formula.
• The process of finding the derivative of any given function is known as differentiation.
• Rules and differentiation formulas help to calculate the derivative of a function and integration.
• Calculus is a branch of mathematics that focuses on limits, functions, derivatives, integrals, and mainly infinite series.
• Integration is also known as anti-derivative.
• Integration determines volume, area, and central point. 

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Previous Year Questions

1. The value of sin⁡² 51^∘+ sin^⁡2 39^∘ is… [KCET – 2020]
2. If cos x = |sin x| then, the general solution is… [KCET – 2019]
3. If 0≤ x< π​/2, then the number of values of x… [JEE Main – 2019]
4. A vertical lamp-post at the midpoint D… [JEE Main – 2019]
5. If tanA+cotA=2, then the value of…  [KCET – 2020]
6. The value of cos²45^∘−sin²15^∘ is… [KCET – 2017]
7. A, B and C are the angles opposite to the corresponding sides of lengths… [JKCET – 2017]
8. The value of tan 8/π​ is equal to… [KCET – 2016]
9. The value of tan⁡10^∘ tan⁡20^∘ tan⁡30^∘ tan⁡40^∘ tan⁡50^∘ tan⁡60^∘… [COMEDK UGET – 2012]
10. A value of θ satisfying sin⁡5θ−sin⁡3θ+sinθ…  [KCET – 2011]