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

Answer 1
By Chain Rule, #y'=1/{2(1+x^2)}#

Let us look at some details.

Let us first find the derivative of #x-sqrt{1+x^2}#. By rewriting the square-root as the 1/2 power, #(x-sqrt{1+x^2})'=[x-(1+x^2)^{1/2}]'#
by Chain Rule, #=1-1/2(1+x^2)^{-1/2}cdot(2x)=1-x/sqrt{1+x^2}#
by taking the common denominator, #={sqrt{1+x^2}-x}/sqrt{1+x^2}#
Now, we can find #y'#. By Chain Rule, #y'=1/{1+(x-sqrt{1+x^2})^2}cdot{sqrt{1+x^2}-x}/sqrt{1+x^2}#
by multiplying out the denominator of the first quotient, #=1/{2(1+x^2-xsqrt{1+x^2})}cdot{sqrt{1+x^2}-x}/sqrt{1+x^2}#
by multiply the quotients together, #={sqrt{1+x^2}-x}/{2[(1+x^2)sqrt{1+x^2}-x(1+x^2)]}#
by factoring out #(1+x^2)# from the denominator, #={sqrt{1+x^2}-x}/{2(1+x^2)(sqrt{1+x^2}-x)}#
by cancelling #sqrt{1+x^2}-x#, #=1/{2(1+x^2)}#
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Answer 2

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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Answer from HIX Tutor

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.

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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