# Can someone explain why this happens?

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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.

You should get the same graph as the original function.

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

Differentiating implicitly gives

or

Simplifying it gives

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I think I understand.

If I graph

I get:

The little blue piece is

For

There is no such restriction on the

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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