# Find the derivative of #sin^2x# using first principles?

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# f'(x) =2sinxcosx #

And so the limit becomes:

And then using the Fundamental trigonometric calculus limits:

we have:

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To find the derivative of (\sin^2(x)) using first principles, we start with the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

For (f(x) = \sin^2(x)), we have:

[ f(x) = \sin^2(x) ] [ f(x + h) = \sin^2(x + h) ]

Now, let's substitute these into the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{\sin^2(x + h) - \sin^2(x)}{h} ]

Now, we can use the identity (\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)):

[ \sin^2(x + h) = (\sin(x + h))^2 = (\sin(x)\cos(h) + \cos(x)\sin(h))^2 ]

[ = \sin^2(x)\cos^2(h) + 2\sin(x)\cos(x)\sin(h)\cos(h) + \cos^2(x)\sin^2(h) ]

[ - \sin^2(x) ]

Now, substitute this expression back into the derivative:

[ f'(x) = \lim_{h \to 0} \frac{\sin^2(x)\cos^2(h) + 2\sin(x)\cos(x)\sin(h)\cos(h) + \cos^2(x)\sin^2(h) - \sin^2(x)}{h} ]

[ = \lim_{h \to 0} \frac{\sin^2(x)\cos^2(h) + 2\sin(x)\cos(x)\sin(h)\cos(h) + \cos^2(x)\sin^2(h) - \sin^2(x)}{h} ]

[ = \lim_{h \to 0} \frac{\sin^2(x)(\cos^2(h) - 1) + 2\sin(x)\cos(x)\sin(h)\cos(h) + \cos^2(x)(\sin^2(h))}{h} ]

[ = \lim_{h \to 0} \frac{\sin^2(x)(\cos^2(h) - 1)}{h} + \lim_{h \to 0} \frac{2\sin(x)\cos(x)\sin(h)\cos(h)}{h} + \lim_{h \to 0} \frac{\cos^2(x)(\sin^2(h))}{h} ]

Now, as (h) approaches 0, (\cos^2(h) - 1) approaches 0, (\sin(h)) approaches 0, and (\sin^2(h)) approaches 0. So:

[ f'(x) = 0 + 0 + 0 = 0 ]

Therefore, the derivative of (\sin^2(x)) using first principles is 0.

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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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