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For each category see if you can devise the rule(s) that was used to find the derivative(s) and then use it to find the derivatives of the functions that follow.

Category 1: Sine and Cosine

1. Function: f(x)=sin?(2x)                 3. Function: f(x)=3sin?(x^2)
    Derivative: f^' (x)=2cos?              (2x) Derivative: f^' (x)=6xcos(x^2)




2. Function: f(x)=cos?(5x)                  4. Function: f(x)=cos?(x^3)
    Derivative: f^' (x)=-5sin?(5x)            Derivative: f^' (x)=-3x^2 sin?(x^3)


What is the formula for the derivative of Sine? What about for Cosine? What role does the chain rule play in these formulas?

Find the derivatives of the following:

f(x)=2sin?(3x-2)    2. f(x)=-cos?(x^2-5x)

Category 2: Tangent and Cotangent

1.Function: f(x)=tan?(3x)                    3. Function: f(x)=7tan?(x^2)
   Derivative: f^' (x)=3?sec?^2 (3x)             Derivative: f^' (x)=14x?sec?^2 (x^2)



2.Function: f(x)=cot?(6x)                      4. Function: f(x)=1/3 cot?(2x^3)
Derivative: f^' (x)=-6?csc?^2 (6x)            Derivative: f^' (x)=-2x^2 ?csc?^2 (2x^3)


What is the formula for the derivative of Tangent? What about for Cotangent?


Find the derivatives of the following:

f(x)=1/2 tan?(x^4)     2. f(x)=6cot?(x^3-2x^2)

Category 3: Secant and Cosecant

1. Function: f(x)=sec?(8x)                   3. Function: f(x)=csc?(9x)
Derivative: f^' (x)=8 sec?(8x) tan?             (8x) Derivative: f^' (x)=-9 csc?(9x) cot?(9x)


2. Function: f(x)=2/5 sec?(3x^5)                              4. Function: f(x)=2csc?(x^6)
Derivative: f^' (x)=6x^4 sec?(3x^5 )tan?(3x^5)              Derivative: f^' (x)=-12x^5 csc?(x^6 )cot?(x^5)


What is the formula for the derivative of Secant? What about for Cosecant?







Find the derivatives of the following:

f(x)=-2sec?(x^2) 2. f(x)=4csc?(2x^3-1)

Combining Rules

If f(x)=xsin(3x) what rule that we have previously learned should we use to find its derivative? Use that rule to find the derivative of f(x).






If f(x)=?tan?^3 (5x) what rule that we have previously learned should we use to find its derivative? Use that rule to find f(x).







If f(x)=(sec?(5x))/x^2 what rule that we have previously learned should we use to find its derivative? Use that rule to find f(x).

Find the derivatives:

f(x)=cos?(2x)sin?(3x) 3. f(x)=?sec?^4 (x^2)







f(x)=(csc?(7x))/(cot?(7x))







Part II: Inverse Trigonometric Functions

Category 1: ArcSine and ArcCosine
Please Note you will also see these functions as ?sin?^(-1) or ?cos?^(-1)

1.Function: f(x)=arcsin?(2x)                       3. Function: f(x)=3sin^(-1)?(x^2)
Derivative: f^' (x)=2/?(1-4x^2 )                Derivative: f^' (x)=6x/?(1-x^4 )




2. Function: f(x)=cos?(5x)                              4. Function: f(x)=cos^(-1) (x^3)
Derivative: f^' (x)=-5/?(1-25x^2 )                   Derivative: f^' (x)=-(3x^2)/?(1-x^6 )


What is the formula for the derivative of ArcSine? What about for ArcCosine?







Find the derivatives of the following:

f(x)=2arcsin?(3x) 2. f(x)=-arccos?(5x^2)





















Category 2: ArcTangent and ArcCotangent
Please note you will also see these functions as ?tan?^(-1) or ?cot?^(-1)

                1.Function: f(x)=arctan?(3x)                      3. Function: f(x)=7tan^(-1)?(x^2)
                Derivative: f^' (x)=3/(1+9x^2 )                  Derivative: f^' (x)=14x/(1+x^4 )



2. Function: f(x)=arccot?(6x)                          4. Function: f(x)=1/3 cot^(-1) (2x^3)
Derivative: f^' (x)=-6/(1+36x^2 )                    Derivative: f^' (x)=-(6x^2)/(3(1+4x^6))



What is the formula for the derivative of ArcTan? What about for ArcCot?







Find the derivatives of the following:

f(x)=1/2 arctan?(x^4) 2. f(x)=6arccot?(?7x?^3)






















Category 3: ArcSecant and ArcCosecant
Please note you will also see these functions as ?sec?^(-1) or ?csc?^(-1)

1. Function: f(x)=arcsec?(8x)                              3. Function: f(x)=arccsc?(9x)
Derivative: f^' (x)=8/(|8x| ?(64x^2-1))                       Derivative: f^' (x)=-9/(|9x| ?(81x^2-1))


2. Function: f(x)=sec^(-1) (3x^5)                           4. Function: f(x)=2csc^(-1) (x^6)
Derivative: f^' (x)= (15x^4)/(|3x^5 | ?(9x^10-1))        Derivative: f^' (x)=-(12x^5)/(|x^6 | ?(x^12-1))


What is the formula for the derivative of ArcSec? What about for ArcCsc?







Find the derivatives of the following:

f(x)=-2arcsec?(x^2) 2. f(x)=4arccsc?(2x^3)

Explanation / Answer

Formula for derivation of arc(sinx)=1/sqrt(1-x^2)

Formula for derivation of arc(cosx)=-1/sqrt(1-x^2)

f(x)=2arcsin?(3x) f'(x)=6/(sqrt(1-9x^2))

f(x)=-arccos?(5x^2) f'(x)=10x/(sqrt(1-25x^4))

formula for the derivative of ArcTan(x)=1/1+x^2

formula for the derivative of Arc cot(x)=-1/1+x^2

f(x)=1/2 arctan?(x^4) f'(x)=2x^3/(1+x^8)

f(x)=6arccot?(?7x?^3) f'(x)=-126x^2/(1+49x^6)

formula for the derivative of ArcSec(x)=1/(mod(x)(sqrt(x^2-1))

formula for the derivative of ArcCosec(x)=-1/(mod(x)(sqrt(x^2-1))

f(x)=-2arcsec?(x^2) f'(x)=-4x/((x^2)(sqrt(x^4-1))

f(x)=4arccsc?(2x^3)   

f'(x)=-12/x(sqrt(4x^6-1) for x>0

f'(x)=12/x(sqrt(4x^6-1) for x<0

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