Question

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

Answer 1

`-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.