# How do you find the first and second derivative of #(lnx)^3#?

f'(x)=

f''(x) =

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To find the first derivative of ( (ln(x))^3 ), we use the chain rule and the derivative of the natural logarithm function:

[ \frac{d}{dx}[(ln(x))^3] = 3(ln(x))^2 \cdot \frac{1}{x} ]

To find the second derivative, we differentiate the result from the first derivative:

[ \frac{d^2}{dx^2}[(ln(x))^3] = \frac{d}{dx}[3(ln(x))^2 \cdot \frac{1}{x}] ]

[ = 3 \cdot 2(ln(x)) \cdot \frac{1}{x} \cdot \frac{1}{x} - 3(ln(x))^2 \cdot \frac{1}{x^2} ]

[ = \frac{6(ln(x))}{x^2} - \frac{3(ln(x))^2}{x^2} ]

[ = \frac{6(ln(x)) - 3(ln(x))^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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