How do you differentiate #f(x)=sin(1/sqrt(3x^2-4) ) # using the chain rule?
1 Answer
Explanation:
Let's break down your function using the chain rule:
#f(x) = sin(color(blue)(1/sqrt(3x^2-4)) ) = sin color(blue)(u)# where
#" "u = 1 / sqrt(color(orange)(3x^2-4)) = 1 / sqrt(color(orange)(v)) # where
#" " v = 3x^2 - 4#
According to the chain rule, the derivative is:
#f'(x) = [sin u]' * u' = [sin u]' * [1 / sqrt(v)]' * v' #
Let's compute the derivatives of those terms!
# [sin u]' = cos u = cos( 1 / (3x^2 - 4))#
#u' = [1 / sqrt(v)]' = [v^(-1/2)]' = -1/2 v^(-3/2) = - 1 / (2 sqrt(v^3)) = - 1 / (2 sqrt((3x^2-4)^3))#
#v' = [3x^2 - 4]' = 6x#
Thus, your derivative is:
#f'(x) = [sin u]' * [1 / sqrt(v)]' * v' #
# = cos( 1 / (3x^2 - 4)) * (- 1 / (2 sqrt((3x^2-4)^3))) * 6x#
# = - cos( 1 / (3x^2 - 4)) * (6x) / (2 sqrt((3x^2-4)^3))#
