Calculus Derivatives of Trig Functions
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Surf's Up!
Derivatives of Trig Functions

 
 The derivatives of trig functions are a bit different.
 Look at the sine function, and the slope of the sine function 
at some selected points.
F(X) = sin X
X slope
0 1
90 0
180 -1
270 0
360 1
 
 Look at these angles and slope values.
 At 0 the value is 1,
 At 90 the value is 0,
 At 180 the value is -1,
 At 270 the value is 0,
 and at 360 the value is back to 1.
 
 That looks like the values of the cosine.
 And it is:

 If F(X) = sinX  then F'(X) = cosX

 
 That's like cosmically weird, but that's the way it works.
 I wonder if the derivative of the cosine might be the sine?
 Let's see...
F(X) = cos X
X slope sinX
0 0 0
90 -1 1
180 0 0
270 1 -1
360 0 0
 
 Well it starts out the same as the sine at zero.
 But when the sine goes positive, the slope of the cosine goes negative.
 And vice a versa
 It looks like the sine but negative ...
 
 BINGO! We have a winner!

  If F(X) = cosX  then F'(X) = - sinX

 
 Not quite what we were expecting, but close.
  

 
 Sometimes we don't just have X as the variable.
 Sometimes we have something more complicated.
 Like 2X.
 OK, that's not so complicated.
 But it's enough.
 
 If we want to find the derivative of F(X) = sin2X, what do we do?
 
 It turns out that this is one of those outside function and inside function things. 
 The outside part is G(X) = sin(something)
 And the inside part is H(X) = 2X    (the something!)
 
 That means to find the derivative we use the power chain rule
 from a couple of pages ago.
 
 So you don't have to go looking, here it is:
 

 If F(X) = H(G(X))  then  F'(X) = H'(G(X)) x G'(X)

 
 So the derivative of the sine is the cosine, and the derivative of 2X is 2.
 
 Putting this all together, and remembering 
that we don't touch the inside function when we do the outside derivative, 
 we get:

 

 If F(X) = sin2X  then F'(X) = cos2X 2  

which simplifies to F'(X) = 2cos2X

 

 
 OK, 4 more trig functions to go.
The deal is, now we're going to cheat.
 
All of the rest of the trig functions can be written as functions of sine and cosine.
Then we just use the rules we already have to get the answers.
 
Here's the tangent:
 
So, use the quotient rule:
 

So the derivative of the tangent is the secant squared.
Let's see what we get for the derivative of the cosecant...
 
 
 Now we have a choice here.
 We can do the derivative as a quotient rule,
 OR
 We can rewrite this as:

 csc X = (sin X)-1

 And do a power chain rule.
 
 We did the quotient rule on the tangent example 
 so lets do the power chain rule on this one.
 
 
 OK, for practice you should stop right now 
and work out the derivatives of the other 3 trig functions.
 
 We'll wait.
 
 Really...
 
 Done already? WOW!
 
 There are three suggestions I have for remembering these 6 derivatives:
 
 1) Don't memorize any of it, but know what the graphs
     of sine and cosine look like.
 Then you can work it all out when you need it. 
 Yes, this will take a good bit of time anytime you need one.
 
 2) Use a whole bunch of brute force memorization and memorize all 6.
 
 3) Memorize 3, and use a trick to know the other three with no extra effort
 (That one sounds like a winner, read on to learn the trick)
 
 THE TRICK
 
 To learn the derivatives of the 6 trig functions 
 you actually only have to learn 3 of them.
 The three to learn are sine, tangent, and secant.
 
IF THEN
F(X) = sin X F'(X) = cos X
F(X) = tan X F'(X) = sec2 X
F(X) = sec X F'(X) = sec X tan X
 
 The other three functions all start with "co"
 Cosine, Cotangent, and Cosecant.
 Think of the functions as having partners:
 
sine to cosine
tangent to cotangent
secant to cosecant
 
 To find the derivative of the "co" functions,
 start with the derivatives of sine, tangent and secant,
 change each function in the derivative to it's co-function partner
 and put a minus sign in front of the derivative.
 
 So starting with one of the three we need to know:
 

 If F(x) = secant X  then F'(x) = secant X tangent X

 
 So now to get the derivative of the cosecant:
 

  If F(X) = cosecant X  then F'(X) = - cosecant X cotangent X

 
 So all 6 trig function derivatives look like this:
 
IF: THEN:
 F(X) = sinX F'(X) = cosX
F(X) = cosX F'(X) = - sinX
F(X) = tanX F'(X) = sec2 X
F(X) = cotX F'(X) = - csc2 X
F(X) = secX F'(X) = sec X tan X
F(X) = cscX F'(X) = -csc X cot X
 

   copyright 2008 Bruce Kirkpatrick

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