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

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Use the product rule to get:

and the same for getting

Step 1. Break the original into two functions

Step 3. Use the Product Rule to solve

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To find the first derivative of ( x(\ln x)^2 ), apply the product rule. The result is ( (2\ln x + 1)(\ln x)^2 + 2x\ln x/x ).

To find the second derivative, differentiate the first derivative obtained above using the product rule and simplify. The result is ( \frac{2}{x}(1 + 2\ln x) + 2(\ln x)^2 + \frac{2}{x} ).

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To find the first and second derivatives of ( x(\ln(x))^2 ), we will use the product rule and the chain rule.

Let ( f(x) = x ) and ( g(x) = (\ln(x))^2 ).

The first derivative of ( f(x) ) is ( f'(x) = 1 ).

The first derivative of ( g(x) ) can be found using the chain rule:

[ g'(x) = 2(\ln(x)) \cdot \frac{1}{x} = \frac{2\ln(x)}{x} ]

Now, applying the product rule:

[ \frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x) ]

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

[ \frac{d}{dx}[x(\ln(x))^2] = (\ln(x))^2 + 2\ln(x) ]

This is the first derivative. Now, let's find the second derivative.

The derivative of ( (\ln(x))^2 ) can be found using the chain rule again:

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

Now, the second derivative using the product rule:

[ \frac{d^2}{dx^2}[x(\ln(x))^2] = \frac{d}{dx}[(\ln(x))^2 + 2\ln(x)] ]

[ \frac{d^2}{dx^2}[x(\ln(x))^2] = 2\cdot \frac{1}{x} + 2\cdot \frac{1}{x} = \frac{2}{x} + \frac{2}{x} = \frac{4}{x} ]

So, the first derivative of ( x(\ln(x))^2 ) is ( (\ln(x))^2 + 2\ln(x) ), and the second derivative is ( \frac{4}{x} ).

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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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