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In Mathematics, transcendental functions are the analytical functions that are not algebraic, and hence do not satisfy the polynomial equation. In other words, transcendental functions cannot be expressed in terms of finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting the roots. The functions such as logarithmic, trigonometric functions , and exponential functions are few examples of transcendental functions.

The transcendental functions can be expressed in algebra only in the terms of infinite sequence. Hence, the term transcendental means non-algebraic.

Transcendental function can be defined as a function that is not algebraic, and cannot be expressed in terms of finite sequence of the algebraic operations such as sin x .

The most familiar transcendental functions examples are the exponential functions, logarithmic functions, trigonometric functions, hyperbolic functions, and inverse of all these functions. The less familiar transcendental functions examples are Gamma, Elliptic, and Zeta functions.

A polynomial equation is an equation in the form of

f(x) = pₙ (x) = a₀ xⁿ⁻¹+ a₁ xⁿ⁻¹ + a₂ xⁿ⁻² +....+aₙ ₋₁ x + aₙ = 0 is known as an algebraic equation.

x⁴ - 4x² - 3 = 0, 4x² - 3x + 9 = 0, 2x³- 5x² - 7x + 3 are some of the algebraic equations.

An equation containing polynomials, logarithmic functions, trigonometric functions , exponential functions is known as transcendental equation.

tan x - eˣ = 0, sin x - xe²ˣ = 0 , xeˣ = cos x are some of the transcendental equations examples.

Transcendental equation is an equation into which transcendental functions (such as exponential, logarithmic, trigonometric, or inverse trigonometric) of one of the variable(s) have been solved for. Transcendental equations do not have closed form solutions.

Transcendental equations examples includes: x =e⁻ ˣ , x = cos x, 2ˣ = x².

**1. Find dy/dx For the Function y = In(tan x + sec x)**

**Solution: **

dy/dx = x² (1/4x. 4) + In (4x). 2x

= x + 2x In ( 4x)

= x( 1 + 2 In (4x))

2. Calculate \[\lim_{x\rightarrow 0}\frac{secx-1}{sinx}\]

Solution:

As both numerator and denominator approaches to 0. Hence, applying L’s hospital rule, we get:

\[\lim_{^{2}x\rightarrow 0}\frac{secx-1}{sinx}\]

\[=\lim_{x\rightarrow 0}\frac{secx-tanx}{cosx}\]

\[=\lim_{x\rightarrow 0}\frac{1.0}{1}\]

= 1

FAQ (Frequently Asked Questions)

1. What are the Transcendental Numbers?

Ans: Transcendental numbers are the numbers that are not considered to be a root of any polynomial with integer coefficients. Transcendental numbers are the opposite of algebraic numbers which are considered as the numbers that are roots of some integer polynomial. The most renowned examples of transcendental numbers are π and e.

2. What are the Different Methods to Find Algebraic and Transcendental Equations?

Ans: The different methods to find algebraic and transcendental equations are:

Bisection Method

Method of False Position

Newton Raphson Method

The Iteration Method

3. Which Functions are Not Considered as Transcendental Functions?

Ans: A function that is not transcendental is known as an algebraic function. Rational functions and square root functions are the examples of transcendental functions. The indefinite integral of many transcendental functions is considered to be transcendental. For example, logarithmic functions are derived from reciprocal functions with a view to find the area of the hyperbolic sector.