Expert: Paul Klarreich - 8/4/2009

Question
) use the composite rule to differentiate the function
f(x)=exp(2/3*sinx) note exp=e^x

2) use the product rule and answer to the q1 to show that the function

g(x)=cosxexp(2/3*sinx) (0<_ x <_ 2pi)

has derivatives

g'(x)=1/3*(2-3sinx-2sin^2x)*exp(2/3*si…

3)find any stationary points of function g(x) defined in q2 and use the first derivatives test to classify each stationary point as a local maximum or minimum of g(x)
domain of the function g(x) is in interval [0,2pi]

can someone please explain step by step if possible

Answer
Questioner: jaymin patel
Country: United Kingdom
Category: Advanced Math
Private: No
Subject: differentiation
Question: use the composite rule

>> called the CHAIN rule here.

to differentiate the function
f(x)=exp(2/3*sinx) note exp=e^x
>> Let y = e^u,  and  u = 2 sin x /3

Then you won't have any trouble getting:

f' = 2/3 cos x  exp(2 sin x / 3)

2) use the product rule and answer to the q1 to show that the function

g(x)= cosx exp(2/3*sinx) (0<_ x <_ 2pi)

>> g(x)= 2/3 cosx exp(2/3*sinx) (0<_ x <_ 2pi)  LOSE THAT 2/3?

>> This is your derivative for part 1, except for a 2/3 factor, which I put in for you.

has derivatives  (derivatives???)

g'(x)= 1/3*(2 - 3sinx - 2sin^2x)*exp(2/3*si…????)

Use  u = cos x,  v = exp(2/3 sin x), and make use of the first part to get the derivative  v'.
    2/3  ---u--- -----v'------     ---u'-- --------v------
g' = 2/3[ (cos x)(exp(2/3*sinx)) + (- sin x)(exp(2/3 sin x)) ]

Factor:

g' = 2/3 exp(2/3*sinx)[cos x - sin x]


3)find any stationary points of function g(x) defined in q2 and use the first derivatives test to classify each stationary point as a local maximum or minimum of g(x)

>> Now just set  cos x - sin x = 0, or  cos x = sin x,  or  tan x = 1,
and solve using your basic trigonometry.


domain of the function g(x) is in interval [0,2pi]