# What is the #n^(th)# derivative of #sin^2x#?

# f^((n)) sin^2x = { (sin^2x,n=0), ((-1)^(n/2+1) \ 2^(n-1) \ cos 2x,n gt 0 " even"), ((-1)^((n+1)/2+1) \ 2^(n-1) \ sin 2x,n gt 0 " odd") :} #

We have:

At first glance we may suspect that to gain further derivatives we will require the product rule and their form will become increasingly more complex. However we note that:

Allowing us to write the first derivative as:

So differentiating further times we get:

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The n^(th) derivative of sin^2(x) is:

[ \frac{{d^n}}{{dx^n}} (\sin^2(x)) = \sin^2(x) + 2n\sin(x)\cos(x) ]

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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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