Find the derivative?

Question

Find the derivative of y = 2sqrt(3 arc tan 5x)

Answer 1

y = 2sqrt(3 arctan 5x)
y/2 = sqrt(3 arctan 5x)
(y^2)/4 = 3 arctan 5x
(y^2)/12 = arctan 5x
tan ((y^2)/12) = 5x
d/dx (tan ((y^2)/12)) = d/dx (5x)
sec^2 ((y^2)/12) (2y/12) (dy/dx) = 5
dy/dx = (30/y) cos^2 ((y^2)/12)
or
y' = (30/2sqrt(3 arctan 5x)) cos^2 (((2sqrt(3 arctan 5x))^2)/12)
y' = (15/sqrt(3 arctan 5x)) cos^2 ((3 arctan 5x)/3)
y' = (15/sqrt(3 arctan 5x)) cos^2 (arctan 5x)^2))
y' = (15/sqrt(3 arctan 5x)) (1/(1 + 25x^2))

Answer 2

y = 2sqrt( 3arctan(5x) )

First, I'm going to convert that sqrt to make it as a power of (1/2) instead.

y = 2 [ 3arctan(5x) ]^(1/2)

Now, I'm going to use the property
(mn)^(1/2) = (m^(1/2))(n^(1/2))

y = 2 (3)^(1/2) [arctan(5x)]^(1/2)

The reason why I did this is that, upon differentiating, constants can be ignored.

To obtain the derivative of this, you have 3 functions to deal with:
d/dx x^(1/2) = (1/2)x^(-1/2)
d/dx arctan(x) = 1/(1 + x^2)
d/dx 5x = 5

Applying the chain rule and various other differentiation rules, we get from here:

y = 2 (3)^(1/2) [arctan(5x)]^(1/2)
to here:
y' = 2 (3)^(1/2) [ (1/2) {arctan(5x)}^(-1/2) ] [1/[1 + (5x)^2] ] [5]

I'm going to merge the constants together and that's it.

y' = 2(3)^(1/2) (1/2) (5) [arctan(5x)]^(-1/2) [1 / (1 + 25x^2)]

y' = 5 (3)^(1/2) [arctan(5x)]^(-1/2) [1 / (1 + 25x^2)]

It doesn't simplify much after that.