Question

What is the derivative of `y=sec(3x^2)`?

Answer 1

By the chain rule:

Let `y = sec(u)` and `u = 3x^2`.

The derivative of `secx` can be found by the following proof:

`secx = 1/cosx`

`(1/cosx)' = ((0 xx cosx) - (1 xx -sinx))/(cosx)^2`

`(1/cosx)' = sinx/(cos^2x)`

`(secx)' = secx xx sinx/cosx`

`(secx)' = secxtanx`

The derivative of `3x^2` can be obtained using the power rule:

`(3x^2)' = 2 xx 3x^(2 - 1)`

`(3x^2)' = 6x`

The chain rule states that `dy/dx = dy/(du) xx (du)/dx`.

Hence, `dy/dx = secutanu xx 6x = 6xsec(3x^2)tan(3x^2)`

Practice exercises:

1. Differentiate the following:

a) `cscx`

b) `cot(3x^2 + 5x + 1)`

c) `tan(e^(2x^2))`

Hopefully this helps, and good luck!