# How do you find the derivative of #(arctan x)^3#?

You can find it like this:

So you apply the chain rule:

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The derivative of (arctan x)^3 with respect to x is 3*(arctan x)^2 / (1 + x^2).

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To find the derivative of ( (\arctan x)^3 ), we can use the chain rule of differentiation. Let's denote ( u = \arctan x ). Then, we have ( y = u^3 ). Using the chain rule, the derivative of ( y ) with respect to ( x ) is:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ]

First, we find ( \frac{dy}{du} ) by applying the power rule to ( u^3 ), which gives us:

[ \frac{dy}{du} = 3u^2 ]

Next, we find ( \frac{du}{dx} ), the derivative of ( \arctan x ) with respect to ( x ). This derivative is ( \frac{1}{1+x^2} ) by the derivative of the arctan function.

Now, we multiply these two derivatives together:

[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot \frac{1}{1+x^2} ]

Finally, substituting ( u = \arctan x ) back into the expression, we get:

[ \frac{dy}{dx} = 3(\arctan x)^2 \cdot \frac{1}{1+x^2} ]

So, the derivative of ( (\arctan x)^3 ) with respect to ( x ) is ( 3(\arctan x)^2 \cdot \frac{1}{1+x^2} ).

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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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