How to find the derivative of cos ( x ) ?

In order to find the derivative of cos ( x ), the fact must be kept in consideration that the derivative of cos ( x ) = 1 / sec ( x ). The steps to solve the derivative of the cos ( x ) is illustrated below.

It should be noticed that 1 / sec ( x ) is a quotient. Therefore, the quotient rule for derivative will be used to find the derivative. The quotient rule for derivatives is as follows.

The Quotient Rule:

The other fact which will be required to find this derivative are as follows:

a ) The derivative of sec ( x ) is { sec (x) tan ( x ) }

b ) The derivative of a constant is 0.

c ) Tan (x) = sin (x) / cos (x)

d ) ( a / b ) / ( c / d ) = ( a / b ) * ( d / c ) = ( a d / b c )

The first thing which will be performed that is the quotient rule on 1 / sec ( x ). In 1 / sec( x ) the function in the numerator is f ( x ) = 1, and the function in the denominator is g ( x ) = sec ( x ). 

According to the quotient rule:

Now, it will be simplified. This is where the fact come into play. During the first part of simplification the first part of simplification will be used on fact 1 and 2.

Now, it will simplified further using the fact 3 and 4

Now it can be putted on all of this work which are nicely organized and compact form as follows

Explain why it is useful to be familiar with the derivatives of basic trigonometric functions. 

The basic trigonometric function include the following 6 functions: sine ( sin x ), cosine ( cos x ), tangent ( tan x ), cotangent ( cot x ), secant ( sec x ) and cosecant ( cosec x ). The derivative helps to find the rate of change, critical point, minimum and maximum value, finding absolute extrema, helps to determine the shape of the graph,’Hospital’s Rule and Indeterminate Forms, Linear Approximations and many more.

The various application of the derivatives of the trigonometric function are explained below in terms of solved examples:

Example 1:

Find the equation of the normal to the curve of  

Solution:  The derivative of  y=arctan(u) is given by:

In this example, we have u =   the derivative is :

When x = 3, this expression is equal to 0.153846

So the slope of the tangent at x = 3 is 0.153846

The slope of the normal at x = 3 is given by

So, the equation of the normal, when x = 3 and y = 0.9828 is given by:

                                                     y=0.982 8= -6.5(x-3)

That is,

    y= -6.5+20.483

Here, the graph of the situation is given by: 

The graph of y=arctan is showing the tangent and the normal at x = 3

Example 2:

The apparent power P_a of an electric circuit whose power is P and whose impedance phase angle is θ, is given by

Pa ​= P sec θ

Given that P is constant at 12 W, find the time rate of change of Pa if θ is changing at the rate of 0.050 rad/min, when θ = 40°.