# What is the derivative of #(arcsin(3x))/x#?

You can solve this problem by using the quotient rule .

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To find the derivative of ( \frac{{\arcsin(3x)}}{x} ), use the quotient rule. The derivative is given by:

[ \frac{d}{dx} \left( \frac{{\arcsin(3x)}}{x} \right) = \frac{x \cdot \frac{d}{dx}(\arcsin(3x)) - \arcsin(3x) \cdot \frac{d}{dx}(x)}{x^2} ]

Using the chain rule, ( \frac{d}{dx}(\arcsin(3x)) = \frac{1}{\sqrt{1-(3x)^2}} \cdot 3 ).

Then ( \frac{d}{dx}(x) = 1 ).

Substitute these into the quotient rule formula:

[ \frac{d}{dx} \left( \frac{{\arcsin(3x)}}{x} \right) = \frac{x \cdot \frac{3}{\sqrt{1-(3x)^2}} - \arcsin(3x) \cdot 1}{x^2} ]

Simplify:

[ \frac{d}{dx} \left( \frac{{\arcsin(3x)}}{x} \right) = \frac{3x}{x \sqrt{1-(3x)^2}} - \frac{\arcsin(3x)}{x^2} ]

[ = \frac{3}{\sqrt{1-(3x)^2}} - \frac{\arcsin(3x)}{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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