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

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To differentiate f(x) = ln((1-x^2)^(-1/2)) using the chain rule, follow these steps:

- Identify the outer function and the inner function.
- Differentiate the outer function with respect to the inner function.
- Multiply by the derivative of the inner function with respect to x.
- Combine the results to find the derivative of the composite function.

Let u = (1-x^2)^(-1/2). Then, f(x) = ln(u).

Now, differentiate ln(u) with respect to u: d(ln(u))/du = 1/u.

Next, differentiate u with respect to x: du/dx = d((1-x^2)^(-1/2))/dx.

Use the chain rule to find du/dx: du/dx = (-1/2)(1-x^2)^(-3/2)(-2x) = x/(1-x^2)^(3/2).

Multiply the derivatives: d(ln(u))/dx = (1/u) * (x/(1-x^2)^(3/2)).

Substitute u back in: d(ln((1-x^2)^(-1/2)))/dx = (1/(1-x^2)^(1/2)) * (x/(1-x^2)^(3/2)).

So, the derivative of f(x) = ln((1-x^2)^(-1/2)) with respect to x using the chain rule is: d(ln((1-x^2)^(-1/2)))/dx = x/(1-x^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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