Derivative for the trig function, Mathematics

Assignment Help:

Derivative for the trig function: We'll begin with finding the derivative of the sine function. To do this we will have to utilize the definition of the derivative. It's been whereas since we've had to utilize this, however sometimes there just isn't anything we can do regarding it.  Following is the definition of the derivative for the sine function.

908_trig function5.png

As we can't just plug in h = 0 to evaluate the limit we will have to use the given trig formula on the first as in the numerator.

sin ( x + h ) = sin ( x ) cos ( h ) + cos ( x ) sin ( h )

Doing this gives us,

10_trig function6.png

As you can see upon by using the trig formula we can combine the first & third term and then factor out sine of that. Then we can break up the fraction in two pieces, both of which can be dealt separately.

Now, here both of the limits are limits as h approaches zero.  In the first limit we contain a sin(x) and in the second limit we contain a cos(x).  Both of these are just functions of x only and as h moves in towards zero it has no affect on the value of x. Thus, as far as the limits are concerned, these two functions are constants & can be factored out of their respective limits.

Doing this gives,

1572_trig function7.png

At this point all we have to do is utilizes the limits in the fact above to finish out this problem.

d (sin ( x )) / dx= sin ( x ) (0) + cos ( x ) (1)= cos ( x )

Differentiating cosine is completed in a similar fashion. It will need a different trig formula, however other than that is an almost identical proof. While done with the proof you should get,

                                           d (cos ( x )) / dx= - sin ( x )

Along with these two out of the way the remaining four are rather simple to get.  Remaining four trig functions can be explained in terms of sine & cosine and these definitions, along with suitable derivative rules, can be utilized to get their derivatives.

Let's take a look at tangent. Tangent is explained as,

                                              tan ( x ) = sin ( x ) /cos ( x )

Now that we have the derivatives of sine & cosine all that we have to do is use the quotient rule on this.  Let's accomplish that.

d ( tan (x ))/ dx = d ( sin ( x ) /cos(x))/dx

                           = cos ( x ) cos ( x ) - sin ( x )(- sin ( x )) /cos ( x ))2

                           = cos2 ( x ) + sin 2 ( x ) /cos2 ( x )

Now, recall that cos2 ( x ) + sin 2 ( x )= 1 and if we also recall the definition of secant in terms of cosine we arrive at,

d ( tan(x))/dx= cos2 ( x ) + sin 2( x ) /cos2 ( x )

                       = 1/cos2 (x )

                       = sec2 ( x )

The remaining three trig functions are also quotients including sine and/or cosine and hence can be differentiated in a same manner.  Following are the derivatives of all six of the trig functions.

Derivatives of the six trig functions

d (sin ( x ))/dx = cos ( x )              d (cos ( x )) /dx = - sin ( x )

d ( tan ( x )) /dx= sec2 ( x )                    d (cot ( x )) /dx= -csc 2 ( x )

d (sec ( x )) = sec (x) tan ( x )           d (csc ( x )) = -csc (x) cot ( x )


Related Discussions:- Derivative for the trig function

Kfso, poijn jjjnjb jbjdbjbj

poijn jjjnjb jbjdbjbj

Algebra, 2x+2y=10 and 3y+4x=9

2x+2y=10 and 3y+4x=9

Hypothesis testing, Hypothesis Testing Definition of Hypothesis Testing...

Hypothesis Testing Definition of Hypothesis Testing - A hypothesis is a claim or an opinion about an issue or item.  Hence it has to be tested statistically in order to esta

Law of Iterative Expectation, #quesSuppose we have a stick of length L. We ...

#quesSuppose we have a stick of length L. We break it once at some point X ~ Unif(0;L). Then we break it again at some point Y ~ Unif(0;X). Use the law of iterated expectation to c

Geometry , Solving for X in isosceles triangles

Solving for X in isosceles triangles

International marketing, what are challenges and solution of international ...

what are challenges and solution of international marketing

Characteristics of time series, Characteristics of Time Series Time se...

Characteristics of Time Series Time series has the given characteristics. a) A long term trend (T) -tendency of the whole series to fall and rise. b) Seasonal variati

How much can they deduct from childcare expenses, A family may deduct 24% o...

A family may deduct 24% of their childcare expenses from their income tax owed. If a family had $1,345 in childcare expenses, how much can they deduct? Find out 24% of $1,345 b

Light take 5.3 × 10-6 seconds calculate standard notation, It takes light 5...

It takes light 5.3 × 10 -6 seconds to travel one mile. What is this time in standard notation? In order to convert this number to standard notation, multiply 5.3 through the f

Math, who created math?

who created math?

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd