# How do you differentiate #f(x) = (1-sqrt(3x-1))^2 # using the chain rule?

We must therefore differentiate both functions.

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To differentiate ( f(x) = (1 - \sqrt{3x - 1})^2 ) using the chain rule, follow these steps:

- Identify the outer function: ( u = (1 - v)^2 ), where ( v = \sqrt{3x - 1} ).
- Differentiate the outer function with respect to its inner variable: ( \frac{du}{dv} = 2(1 - v) ).
- Identify the inner function: ( v = \sqrt{3x - 1} ).
- Differentiate the inner function with respect to ( x ): ( \frac{dv}{dx} = \frac{1}{2\sqrt{3x - 1}} ).
- Apply the chain rule: ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} ).
- Substitute the expressions for ( \frac{du}{dv} ) and ( \frac{dv}{dx} ) into the chain rule formula.
- Simplify the expression.

( \frac{df}{dx} = 2(1 - \sqrt{3x - 1}) \cdot \frac{1}{2\sqrt{3x - 1}} )

( \frac{df}{dx} = \frac{1 - \sqrt{3x - 1}}{\sqrt{3x - 1}} )

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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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