Directions: For each category see if you can devise the rule(s) that was used to

Question

Directions: 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

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


1)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:

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















Category 2: Tangent and Cotangent 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.

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)



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:

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
























Category 3: Secant and Cosecant 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.

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:

1. 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

Explanation / Answer

Function: f(x)=sin?(2x) - Derivative: f^' (x)=2cos?(2x)

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

here we find when the function is like sin or cos then we got the rule

i.e.., for sin derivative is cos and for cos dervative is -sin

and the we should multiple the derivative of inside function of sin or cos ---- this is the rule

derivatives of 1. f(x)=2sin?(3x-2) is---- f''(x)= 2(3)cos (3x-2)=6cos(3x-2)

2.f(x)=-cos?(x^2-5x) is f''(x)=-(2x-5)*-sin(x^2-5x)=(2x-5)sin(x^2-5x)

Category 2:

Function: f(x)=tan?(3x)   Derivative: f^' (x)=3?sec?^2 (3x)

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

for this rule is for tan dervative is secant

and simmilarly secant * derivative of interior function this is the rule

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

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

arc means inverse of the function dervative of arc(cotx ) is 1/(1+x^2)

and here also simmalarly product with interior function

dervative of 1.f(x)=1/2 arctan?(x^4) is 1/2*(4X^3)/(1-x^8)===2x^3/(1-x^8)

2. f(x)=6arccot?(?7x?^3) is 6*21x^2/(1+49x^6)==126x^2/(1+49x^6)

Category 3:

Function: f(x)=arcsec?(8x) Derivative: f^' (x)=8/(|8x| ?(64x^2-1))

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

it clearly says that derivative of arc(secx) is 1/ mod(x)*sqrt(x^2-1)

   Function: f(x)=arccsc?(9x) Derivative: f^' (x)=-9/(|9x| ?(81x^2-1))

. Function: f(x)=2csc^(-1) (x^6)   Derivative: f^' (x)=-(12x^5)/(|x^6 | ?(x^12-1))

here it clearly says that dervative of arc(cscx) is -1/mod(x)*sqrt(x^2-1)

dervatives of 1. f(x)=-2arcsec?(x^2) is f"(x)=-4x/mod(x)*sqrt(x^4-1)

2.f(x)=4arccsc?(2x^3) is f'(x)=-24x^2/mod(x)*sqrt(4x^6-1)

this are the rules

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