# How do you find the first and second derivative of #ln(x^(1/2))#?

For the first derivative you can use the chain rule:

but you can also observe that:

Either way the second derivative is:

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To find the first derivative of ln(x^(1/2)), you apply the chain rule, which states that the derivative of ln(u) is (1/u) * u', where u' is the derivative of u with respect to x. Here, u = x^(1/2). Thus, the first derivative of ln(x^(1/2)) is (1/(x^(1/2))) * (1/2) * x^(-1/2), which simplifies to (1/(2x)).

To find the second derivative, you differentiate the first derivative with respect to x. The derivative of (1/(2x)) is -1/(2x^2). So, the second derivative of ln(x^(1/2)) is -1/(2x^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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