Differentiation of Trigonometric

Introduction

Inverse Trigonometric functions are also known as Arcus functions, anti-trigonometric functions, and cyclometric functions. This mathematical function is applied to gain angles for a given trigonometric function set. Inverse trigonometric functions have several applications in geometry, navigation, and engineering. If the different functions can be constructed to map every reference point or target back to its reference, then the function obtained would be the inverse function (f). Some functions include the Real-valued functions. Every target has a specific reference point.

What are the derivatives of inverse trigonometric functions?

In a given function (f) defined on an interval (I), f is used to plot x to f(x). For referring, x is used, and f(x) is the reference point of x. 

Conjointly, the set of all reference points of (f) under (I) forms the set commonly known as the image of (f) under (I), known as f(I). These terms of the function (f) are plotted from (I) to f(I).

And, if the different functions can be made or constructed to map every reference point or target back to its reference, then the function obtained would be the inverse function (f) or (f^-1). Not every function in calculus has an inverse function, but a limited class of functions is injective. Injective functions are the function that has their reference point located to a different target.

Every target has a specific reference point, which concludes that it would be possible to define a function, which plots each target in f(I) back to its initial position (I). These functions are known as the Inverse function of (f).

The functions include the Real-valued functions or include Real numbers, which mostly have their domains as the union of open and closed intervals. If we see these Real valued functions in the context of calculus, then the function is mostly known as Differentiable and continuous functions. 

These Invertible functions have a series of fundamental and useful theorems, which are:

Theorem 1– Invertible functions are also Injective Functions.

A function f is defined on an interval I, which has a large enough domain and, if the function is injective on (I), then f is the domain limited to I have an inverse which is f^-1 having domain f(I).

Theorem 2– Continuous and Invertible functions are Monotone.

A function f is defined on an interval I (mostly having enough large domain). If the function is injective and continuous on I, then f is said to be strictly decreasing or strictly increasing on (I).

Theorem 3– Continuous function lines up with Continuous Inverse.

A function f is defined on an interval I(mostly having enough large domain), f is injective and continuous on I, then f^-1, which is the inverse of f, is defined on f(I) and is continuous on f(I) too. And, f^-1 is strictly decreasing on f(I), and f is increasing on (I).

Derivative of Inverse trigonometric Functions

Several mathematicians had noticed the connection between a point in a function and its relation in inverse functions. Distinctly, it turns out that the slopes of tangent lines at the two points are present, and both are exactly reciprocal to each other. 

By studying calculus, this geometrical second sight would then consist of the backbone known as the inverse function theorem, which can be proved with the help of the three theorems mentioned above.

Applications of the inverse trigonometric functions theorem

As we all know, the differential and inverse functions mostly occur in pairs, so one can use the inverse function theorem to conclude the derivative from the other.

• Linear Functions

So let the original function be F(x)= 7x– 5, then we had solved for x and interchanged the x with F(x), in result we get F^-1(x) = x+5/7, resulting that F’(x)= 7 and (F^-1)’(x) = 1/7

Hence, in the above example, we just found the derivatives are reciprocal of each other.

• Square root Functions

We know that the square root and square functions are inverse, limiting their domain to f(x) = x^2. It should be an easy task to calculate the square root function value with the help of inverse.

Let’s take an example;

f^-1= square root of x at 5;

f^-1 = square root of 5,

So f^-1(x) = square root of x as square root of 5 is to f(x)= x^2

f’(square root of 5) = [2x] with base x= square root 5 = 2(square root 5),

Its reciprocal = ½(square root of 5) = (square root of x)’(5).

• General Roots

We know that x^n and n(square root of x) is inversely proportional to each other(n belongs to N, n> equals to 2), it should still be mentioned that there are only two types of root functions which are odd root functions and even root functions.

The domain of x^n and n(square root of x) is f(x)= x^n is limited to non-negative numbers, but their root is odd, so the domain of these two inverse functions is the full domain of f(i.e., R).

Conclusion

Every function in calculus has an inverse function, but a limited class of functions is injective. Injective functions are the function that has their reference point located to a different target. It turns out that the slopes of tangent lines at the two points are present, and both are exactly reciprocal to each other. There are six trigonometric functions in mathematics: sine, cosine, tan, cosec, cot, and sec. In this article, we have discussed the derivative of trigonometric functions.