# How do you Find the derivative of #y=arctan(x-sqrt(1+x^2))#?

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To find the derivative of ( y = \arctan(x - \sqrt{1 + x^2}) ), we will use the chain rule of differentiation.

Let ( u = x - \sqrt{1 + x^2} ). Then, ( \frac{du}{dx} = 1 - \frac{1}{2\sqrt{1 + x^2}} \cdot 2x = 1 - \frac{x}{\sqrt{1 + x^2}} ).

Now, using the chain rule: [ \frac{dy}{dx} = \frac{d}{dx}(\arctan(u)) = \frac{d(\arctan(u))}{du} \cdot \frac{du}{dx} = \frac{1}{1+u^2} \cdot \left(1 - \frac{x}{\sqrt{1 + x^2}}\right) ]

Substitute back ( u = x - \sqrt{1 + x^2} ) to get the final derivative: [ \frac{dy}{dx} = \frac{1}{1+(x - \sqrt{1 + x^2})^2} \cdot \left(1 - \frac{x}{\sqrt{1 + x^2}}\right) ]

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