Cracking The Ap Calculus ab Exam 2018

SOLUTIONS TO PRACTICE PROBLEM SET 6

1. 2sin x cos x or sin 2x

Recall  that        =   (sin    x)  =   cosx.   Here    we  use the Chain   Rule    to  find    the derivative:     =   2(sin

x)(cos  x). If  you recall  your    trigonometric   identities, 2sin    x   cos x   =   sin 2x. Either  answer  is

acceptable.

2. −2xsin(x^2 )

Recall   that    (cos   x)   =   −sinx.  Here,   we  use     the     Chain   Rule    to  find    the     derivative:     =

(−sin(x^2 ))(2x)    =   −2xsin(x^2 ).

3. 2 sec^3 x−sec x

Recall  that     (tan   x)  =   sec^2 x and that     (sec   x)  =   sec x   tan x.  Using   the Product Rule,   we  get 

    =   (tan    x)(sec  x   tan x)  +   (sec    x)(sec^2 x).    This    can be  simplified  to  sec^3 x +   sec x   tan^2 x =   2

sec^3 x −   sec x.

4.

Recall  that     (sin   x)  =   cos x.  Here    we  use the Chain   Rule    to  find    the derivative:     =    (sin

3 x) (cos   3x)(3). This    can be  simplified  to      =   .

5.

Recall  that     (sin   x)  =   cos x.  Here    we  use the Quotient    Rule    to  find    the derivative:     =   

. This can be simplified to .
6. −4x csc^2 (x^2 ) cot(x^2 )

Recall  that     (csc   x)  =   −csc    x   cot x.  Here    we  use the Chain   Rule    to  find    the derivative:     =