Question
give an example of a function where the derivative does not exist everywhere in its domain
Answers
GPT 3.5
One example of a function where the derivative does not exist everywhere in its domain is the absolute value function.
Consider the function f(x) = |x|, which gives the absolute value of x for any real number x.
For x > 0, the function f(x) = x, and its derivative is f'(x) = 1.
For x < 0, the function f(x) = -x, and its derivative is f'(x) = -1.
However, at x = 0, the derivative of f(x) does not exist because the slope of the function changes abruptly from -1 to 1 at that point.
Thus, the derivative of the absolute value function does not exist at x = 0, making it an example of a function where the derivative does not exist everywhere in its domain.
Consider the function f(x) = |x|, which gives the absolute value of x for any real number x.
For x > 0, the function f(x) = x, and its derivative is f'(x) = 1.
For x < 0, the function f(x) = -x, and its derivative is f'(x) = -1.
However, at x = 0, the derivative of f(x) does not exist because the slope of the function changes abruptly from -1 to 1 at that point.
Thus, the derivative of the absolute value function does not exist at x = 0, making it an example of a function where the derivative does not exist everywhere in its domain.