Trigonometry is the branch of Mathematics that has made itself indispensable for other branches of higher Mathematics may it be calculus, vectors, three dimensional geometry, functions-harmonic and simple and otherwise just can not be processed without encountering trigonometric functions. Further within the specific limit, trigonometric functions give us the inverses as well.

The question now arises: Are all the rules of finding the derivative studied by us so far appliacable to trigonometric functions?

This is what we propose to explore in this lesson and in the process, develop the fornulae or results for finding the derivatives of trigonometric functions and their inverses. In all discussions involving the trignometric functions and their inverses, radian measure is used, unless otherwise specifically mentioned.

OBJECTIVES

**After seen this video, you will be able to:**

- find the derivative of trigonometric functions from first principle;

- find the derivative of inverse trigomometric functions from first principle;

- apply product, quotient and chain rule in finding derivatives of trigonometric and inversen trigonometric functions;

- find and find second order derivative of a functions.

EXPECTED BACKGROUND KNOWLEDGE

- Knowledge of trigonometric ratios as functions of angles.

- Standard limits of trigonometric functions.

- Definition of derivative, and rules of finding derivatives of function.

DERIVATIVE OF TRIGONOMETRIC FUNCTIONS FROM FIRST PRINCIPLE

(i) Let y = sin x

CHECK YOUR PROGRESS 27.1

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

You heve learnt how we can find the derivative of a trigonometric function from first principle and also how to deal with these functions as a function of a function as shown in the alternative method. Now we consider some more examples of these derivatives.

CHECK YOUR PROGRESS 27.2

DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS FROM FIRST PRINCIPLE

CHECK YOUR PROGRESS 27.3

DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS

In the previous section, we have learnt to find derivatives of inverse trignometric functions by first principle. Now we learn to find derivatives of inverse trigonometric functions using these results.

** Example 27.13** :

**Find the derivative of each of the following:**

*Solution:*

*Example 27.14 *: Find the derivative of each of the following:

*Solution:*

*Solution:*

CHECK YOUR PROGRESS 27.4

**Find the derivative of each of the following functions w.r.t. x and express the result in the simplest form (1-3):**

SECOND ORDER DERIVATIVES

We know that the second order derivative of a functions is the derivative of the first derivative of that function. In this section, we shall find the second order derivatives of trigonometric and inverse trigonometric functions. In the process, we shall be using product rule, quotient rule and chain rule.

Let us take some examples.

** Example 27.16:** Find the second order derivative of

*Solution:*