Question

What is the derivative of this function `y=sec^-1(e^(2x))`?

Answer 1

`(2)/(sqrt(e^(4x)-1)`

Explanation:

As if `y=sec^-1x` the derivative is equel to `1/(xsqrt(x^2-1))`
so by using this formula and if `y=e^(2x)` then derivative is `2e^(2x)` so by using this relation in the formula we get the required answer. as `e^(2x)` is a function other than `x` that is why we need further derivative of `e^(2x)`

Answer 2

`2/(sqrt(e^(4x)-1))`

Explanation:

We have `d/dxsec^-1(e^(2x))`.

We can apply the chain rule, which states that for a function `f(u)`, its derivative is `(df)/(du)*(du)/dx`.

Here, `f=sec^-1(u)`, and `u=e^(2x)`.

`d/dxsec^-1(u)=1/(sqrt(u^2)sqrt(u^2-1))`. This is a common derivative.

`d/dxe^(2x)`. Chain rule again, here `f=e^u` and `x=2x`. The derivative of `e^u` is `e^u`, and the derivative of `2x` is `2`.

But here, `u=2x`, and so we finally have `2e^(2x)`.

So `d/dxe^(2x)=2e^(2x)`.

Now we have:

`(2e^(2x))/(sqrt(u^2)sqrt(u^2-1))`, but since `u=e^(2x)`, we have:

`(2e^(2x))/(sqrt((e^(2x))^2)sqrt((e^(2x))^2-1))`

`(2e^(2x))/(e^(2x)sqrt((e^(4x))-1))`

`2/(sqrt(e^(4x)-1))`, our derivative.