If f(x) = ln(x) for x > 0, is f differentiable at x = 1?
Practice Questions
Q1
If f(x) = ln(x) for x > 0, is f differentiable at x = 1?
Yes
No
Only continuous
Only left differentiable
Questions & Step-by-Step Solutions
If f(x) = ln(x) for x > 0, is f differentiable at x = 1?
Step 1: Identify the function f(x) = ln(x). This function is defined for x > 0.
Step 2: Determine the point of interest, which is x = 1.
Step 3: Find the derivative of the function f(x). The derivative f'(x) is calculated as 1/x.
Step 4: Substitute x = 1 into the derivative to find f'(1). This gives f'(1) = 1/1 = 1.
Step 5: Since the derivative f'(1) exists and is equal to 1, we conclude that f is differentiable at x = 1.
Differentiability – The question tests the understanding of whether a function is differentiable at a specific point, which involves checking the existence of the derivative at that point.
Natural Logarithm Function – The function f(x) = ln(x) is a common function in calculus, and its properties, such as continuity and differentiability, are important to understand.
