# How do you find the derivative of #(sqrt(x^2-x+1)-1)/(1+ sqrtx)#?

You find it very carefully...lol

Use the Quotient Rule, Power Rule, Linearity, and the Chain Rule:

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To find the derivative of the given function, apply the quotient rule. The quotient rule states that if you have a function in the form of f(x)/g(x), where f(x) and g(x) are differentiable functions, the derivative is (g(x)*f'(x) - f(x)*g'(x)) / (g(x))^2.

Let f(x) = sqrt(x^2 - x + 1) - 1 and g(x) = 1 + sqrt(x).

Now, differentiate f(x) and g(x) separately to find f'(x) and g'(x). Then, apply the quotient rule using these derivatives.

f'(x) = (1/2)*(x^2 - x + 1)^(-1/2)*(2x - 1) = (x - 1) / (2*sqrt(x^2 - x + 1))

g'(x) = (1/2)*(x)^(-1/2) = (1 / (2*sqrt(x))

Now, apply the quotient rule:

f'(x) = (g(x)*f'(x) - f(x) g'(x)) / (g(x))^2
= [(1 + sqrt(x))(x - 1) / (2*sqrt(x^2 - x + 1))] - [(sqrt(x^2 - x + 1) - 1)

*(1 / (2*sqrt(x))) / (1 + sqrt(x))^2

Simplify the expression as needed.

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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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