Question

What is `d/(d theta) (sec^2 4theta)` ?

Answer 1

`d/(d theta) (sec^2 4theta) = 8sec^2 4theta tan 4theta`

Explanation:

`d/(d theta) (sec^2 4theta) = 2sec 4theta * d/(d theta) (sec 4 theta)` (Power rule and Chain rule)

`= 2sec 4theta * tan 4theta sec 4theta * 4` (Standard differential and Chain rule)

`= 8 sec^2 4theta tan 4theta`

Answer 2

`8sec^2\4thetatan4theta`

Explanation:

Given: `d/(d\theta)(sec^2\4theta)`

We use the chain rule, which states that:

`dy/dx=dy/(du)*(du)/dx`

In this case, it's:

`dy/(d\theta)=dy/(du)*(du)/(d\theta)`

And so, let `u=4theta`, then `du=4 \ d\theta,(du)/(d\theta)=4`.

Also, `y=sec^2u`, then `dy/(du)=2sec^2utanu`.

Therefore, we get:

`dy/dx=2sec^2utanu*4`

`=8sec^2utanu`

Reversing back our substitution, we get,

`=8sec^2\4thetatan4theta`