# What is the derivative of this function #sin(x) / (1 + sin^2(x))#?

You would apply the rule of division:

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To find the derivative of the function ( \frac{\sin(x)}{1 + \sin^2(x)} ), you can use the quotient rule. Let ( u(x) = \sin(x) ) and ( v(x) = 1 + \sin^2(x) ). Then, apply the quotient rule:

[ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - v'(x)u(x)}{v(x)^2} ]

The derivative of ( u(x) = \sin(x) ) is ( u'(x) = \cos(x) ), and the derivative of ( v(x) = 1 + \sin^2(x) ) is ( v'(x) = 2\sin(x)\cos(x) ).

Now, plug these into the quotient rule formula:

[ \frac{d}{dx} \left( \frac{\sin(x)}{1 + \sin^2(x)} \right) = \frac{\cos(x)(1 + \sin^2(x)) - 2\sin(x)\cos(x)\sin(x)}{(1 + \sin^2(x))^2} ]

Simplify this expression to get the derivative of the function.

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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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