How do you use the chain rule to differentiate #f(x)=37-sec^3(2x)#?

1 Answer
Apr 10, 2018

#-6tan(2x)sec^3(2x)#

Explanation:

To differentiate this function, we can ignore the constant part, which is 37, because it will become zero after differentiation.

Let's focus on #-sec^3(2x)#.
You might want to rewrite it as #-(sec(2x))^3# for better conceptualisation when applying the chain rule.

Then, we can further simplify it by substituting a variable for the secant part:
#-u^3#

We get #-3u^2u'# from that. #u'# represents the derivative of #sec(2x)#, which is #2sec(2x)tan(2x)#, number '2' comes from the #2x# part as its derivative.

What we have:
#u=sec(2x)#
#u'=2sec(2x)tan(2x)#

Substitute them back to #-3u^2u'#, we have #-3sec^2(2x)*2sec(2x)tan(2x)#.

Finally, clear up the expression by combining like terms.
#-6tan(2x)sec^3(2x)#

The method I used is called u-sub, it will come handier when you deal with the integrals. It helps you avoid losing yourself in the process when handling with long chains. Of course, if you can keep a neat scratch, the u-sub is not necessary for this type of chain rule functions.

Hope it helps.