# How do you find the derivative of #ln[sqrt(2+x^2)/(2-x^2)]#?

see below

Use the following Properties of Logarithms to expand the problem first before finding the derivative

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To find the derivative of ln[sqrt(2+x^2)/(2-x^2)], you would use the chain rule and the quotient rule. The derivative of ln(u) is (1/u) * u', where u is a function of x. Applying the chain rule, the derivative of sqrt(2+x^2) with respect to x is (1/2)* (2+x^2)^(-1/2) * (2x). The derivative of (2-x^2) with respect to x is -2x.

Using the quotient rule, the derivative of (sqrt(2+x^2)/(2-x^2)) is [(2-x^2) * (1/2)*(2+x^2)^(-1/2) * (2x) - sqrt(2+x^2) * (-2x)] / (2-x^2)^2.

Therefore, the derivative of ln[sqrt(2+x^2)/(2-x^2)] with respect to x is:

[(2-x^2) * (1/2)*(2+x^2)^(-1/2) * (2x) - sqrt(2+x^2) * (-2x)] / [(2-x^2)^2 * sqrt(2+x^2)/(2-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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