How do you differentiate #f(x)=cos^2(1/(3x-1))# using the chain rule?
1 Answer
Explanation:
Let's define the chain of compositions:
#f(x) = cos^2(1/(3x-1)) = [color(orange)(cos(1/(3x-1)))]^2 = color(orange)(u)^2 = u^2#
where
#u = cos(1/(3x-1)) = cos(color(blue)(1/(3x-1))) = cos(color(blue)(v)) = cos(v) #
where
#v = 1 / (3x - 1) = 1 / color(red)(3x-1) = 1 / color(red)(w) = 1/w#
where
#w = 3x-1#
Now, you need to compute the derivatives of those four functions:
#[u^2]' = 2u = 2cos(1/(3x-1))#
#u' = [cos v]' = - sin v = - sin( 1 / (3x - 1))#
#v' = [1/w]' = [w^(-1)]' = -w^(-2) = - 1 /w^2 = - 1 /(3x-1)^2#
#w'= [3x - 1]' = 3#
The derivative of
#f'(x) = [u^2]' * u' * v' * w' #
#= 2cos(1/(3x-1)) * (- sin( 1 / (3x - 1))) * (- 1 /(3x-1)^2) * 3 #
#= 6cos(1/(3x-1)) * sin( 1 / (3x - 1)) * 1 /(3x-1)^2 #