The function f(x) = ln(x) is differentiable at x = 1?

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Question: The function f(x) = ln(x) is differentiable at x = 1?

Options:

Correct Answer: Yes

Solution:

f\'(x) = 1/x; f\'(1) = 1/1 = 1, hence it is differentiable at x = 1.

The function f(x) = ln(x) is differentiable at x = 1?

The function f(x) = ln(x) is differentiable at x = 1?

Step 1: Identify the function we are working with, which is f(x) = ln(x).
Step 2: Understand that we need to find the derivative of the function to check if it is differentiable at x = 1.
Step 3: Use the derivative formula for the natural logarithm, which is f'(x) = 1/x.
Step 4: Substitute x = 1 into the derivative formula: f'(1) = 1/1.
Step 5: Calculate the result: f'(1) = 1.
Step 6: Since the derivative exists and is equal to 1 at x = 1, we conclude that the function f(x) = ln(x) is differentiable at x = 1.

Differentiability – The ability of a function to have a derivative at a given point.
Natural Logarithm Function – The function f(x) = ln(x) is defined for x > 0 and is differentiable in its domain.
Derivative Calculation – Finding the derivative of a function and evaluating it at a specific point.