# Can someone explain why this happens?

##
Say we have a function of #x# , for example #y=x^(1/2)# , we can also write this as #x=y^2#

By taking the derivative of #y=x^(1/2)# , we get #y'=x^(-1/2)/2#

By implicity differenciating #y^2=x# we get #y'=1/(2y)#

If we make #1/(2y)=x^(-1/2)/2# and rearrange to make #y# the subject, we get #y=x^(1/2)# , which was the original function.

However, for other functions, i.e. #y=sqrt(arcsin(x))# the function for equating both differentials gives a graph which include #y=sqrt(arcsin(x))# along with some other stuff.

Say we have a function of

By taking the derivative of

By implicity differenciating

If we make

However, for other functions, i.e.

##### 2 Answers

You should get the same graph as the original function.

#### Explanation:

If

#{dy}/{dx} = 1/(2sqrt((1-x^2)arcsin(x)))#

On the other hand, we can write the first equation as

#x = sin(y^2)#

Differentiating implicitly gives

#1 = cos(y^2) (2y) {dy}/{dx}#

or

#{dy}/{dx} = 1/(2ycos(y^2))#

If you want to equate the 2 "different"

#1/(2sqrt((1-x^2)arcsin(x))) = 1/(2ycos(y^2))#

Simplifying it gives

#(1-x^2)arcsin(x) = y^2cos^2(y^2)#

which seem complicated but notice that the original equation

I think I understand.

#### Explanation:

If I graph

I get:

The little blue piece is

For

There is no such restriction on the