How do you differentiate #f(x)=sec(1/x^3)# using the chain rule?

1 Answer
Nov 29, 2016

Answer:

#f'(x)=-3/x^4sec(1/x^3)tan(1/x^3)#

Explanation:

Differentiating this function is determined by applying chain rule.
#" "#
Chain Rule:
#" "#
#color (blue)((h(g(x)))'=h'(g(x))xxg'(x))#
#" "#
Applying the chain rule :
#" "#
#(sec(1/x^3))'=sec'(1/x^3)xx (1/x^3)'#
#" "#
Knowing that:
#""#
#d/dx(secx)=tanxsecx#
#" "#
#sec'(1/x^3)=tan(1/x^3)xxsec(1/x^3)#
#" "#
#1/x^3" "# can be differentiated by either applying quotient or power rule, here I will apply the power rule.
#" "#
#(1/x^3)'=x^(-3)=-3x^(-4)=-3/x^4#
#" "#
Therefore,
#" "#
#(sec(1/x^3))'=sec'(1/x^3)xx (1/x^3)'#
#" "#
#(sec(1/x^3))'=sec(1/x^3)xxtan (1/x^3)xx(-3/x^4)#
#" "#
Hence,
#" "#
#f'(x)=-3/x^4sec(1/x^3)tan(1/x^3)#