# How do you sketch the graph #y=ln(1/x)# using the first and second derivatives?

Using the properties of logarithms we should see that:

Then we calculate the first and second derivatives:

graph{ln(1/x) [-10, 10, -5, 5]}

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To sketch the graph of (y = \ln(1/x)) using the first and second derivatives, follow these steps:

- Find the first derivative, (y' = \frac{d}{dx}[\ln(1/x)]).
- Simplify the first derivative.
- Find the second derivative, (y'' = \frac{d^2}{dx^2}[\ln(1/x)]).
- Determine the critical points where the first derivative is zero or undefined.
- Use the first and second derivatives to determine the behavior of the graph around critical points.
- Sketch the graph based on the information gathered.

The first derivative is (y' = -\frac{1}{x}) and the second derivative is (y'' = \frac{1}{x^2}).

The critical point occurs at (x = 0).

The first derivative is negative for (x > 0) and positive for (x < 0). The second derivative is positive for all (x), indicating a concave up graph.

The graph approaches the x-axis as (x) approaches infinity, and it approaches the line (y = 0) as (x) approaches zero from either side.

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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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- How do you graph the function #y=5x-x^2#?
- How many points of inflection does the function #f(x) = (pi/3)^((x^3)-8)# have?

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