The function f(x) = e^x is differentiable at all points?

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Question: The function f(x) = e^x is differentiable at all points?

Options:

Correct Answer: True

Solution:

f(x) = e^x is differentiable everywhere as it is an exponential function.

The function f(x) = e^x is differentiable at all points?

The function f(x) = e^x is differentiable at all points?

Step 1: Understand what differentiable means. A function is differentiable at a point if it has a defined slope (derivative) at that point.
Step 2: Identify the function given in the question, which is f(x) = e^x.
Step 3: Recognize that e^x is an exponential function. Exponential functions are known to be smooth and continuous.
Step 4: Check if there are any points where the function might not be differentiable. Since e^x is defined for all real numbers, there are no such points.
Step 5: Conclude that since e^x is smooth and continuous everywhere, it is differentiable at all points.

Differentiability of Exponential Functions – Exponential functions, such as f(x) = e^x, are continuous and differentiable at all points in their domain.