# How do I find the derivative of #y = arccos((x-3)^2)#?

from the formula

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To find the derivative of ( y = \arccos((x-3)^2) ), you can use the chain rule. Here's the step-by-step process:

- Let ( u = (x-3)^2 ).
- Find ( \frac{du}{dx} ).
- Differentiate ( \arccos(u) ) with respect to ( u ), which gives ( \frac{d}{du}(\arccos(u)) = -\frac{1}{\sqrt{1-u^2}} ).
- Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ).
- Substitute ( \frac{dy}{du} ) and ( \frac{du}{dx} ) into the chain rule expression.
- Simplify the expression.

Applying these steps:

- Let ( u = (x-3)^2 ).
- Find ( \frac{du}{dx} ): ( \frac{du}{dx} = 2(x-3) ).
- Differentiate ( \arccos(u) ) with respect to ( u ): ( \frac{d}{du}(\arccos(u)) = -\frac{1}{\sqrt{1-u^2}} ).
- Apply the chain rule: ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} ).
- Substitute ( \frac{dy}{du} ) and ( \frac{du}{dx} ) into the chain rule expression: ( \frac{dy}{dx} = -\frac{1}{\sqrt{1-((x-3)^2)^2}} \cdot 2(x-3) ).
- Simplify the expression.

So, the derivative of ( y ) with respect to ( x ) is ( -\frac{2(x-3)}{\sqrt{1-((x-3)^2)^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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