# What is the #n^(th)# derivative of #x^(-1/2) #?

I hope I'm interpreting this correctly.

Then,

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# y^((n)) = (-1)^n \ ( (2n)! ) / ( (4^n) \ n!) x^(-(2n+1)/2) \ \ \ n in NN #

This can readily be formed by repeated application of the power rule, viz:

And differentiating again we get:

And differentiating again we get:

And differentiating again we get:

And we can see a clear pattern forming, and that:

We can simplify this further as we not the product if the odd consecutive integers can be written as:

Hence, we have:

A formal proof can readily be established via induction, should it be required:

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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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