For the function below: (i) What is the domain of the function? (ii) Find the derivatives of the function. (iii) What is the domain of the derivative?

Question

a) h(u) = tan^-1(u³)

b) y = arcsin (e^x/1+e^x)

Answer 1

To find the domain of a function, start by visualizing the behavior at extreme values and common reference points (eg where a variable is zero)

In the first equation:

The tangent of an angle varies from 0 to ±∞, so it appears that the domain for u will be "all real numbers".

The derivative of tan^-1 (x) is 1/(1 + x²). To get the derivative of tan^-1 (u³ ), use the chain rule:

h(u) = tan^-1(u³)

h'(u) = 1/(1 + (u³)²) ) * (3u²)

simplify:

h'(u) = 3u² / (1 + u⁶)

To find the domain, try some key values: There are no issues with u = 0 or very large values (+ or -).

Since u⁶ cannot be negative, the equation never "blows ups", so it appears that the domain for h' is also "all real numbers".

To visualize things like this, use an online plotting program such as DESMOS:

https://www.desmos.com/calculator