The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the derivative of f(x) = |x| at x = 0?
Solution: The left-hand derivative is -1 and the right-hand derivative is 1, hence the derivative at x = 0 is undefined.
Steps: 7
Step 1: Understand what a derivative is. The derivative of a function at a point gives the slope of the tangent line to the function at that point.
Step 2: Identify the function we are working with, which is f(x) = |x|. This function represents the absolute value of x.
Step 3: Recognize that the function |x| has different behaviors on either side of x = 0. For x < 0, |x| = -x, and for x > 0, |x| = x.
Step 4: Calculate the left-hand derivative at x = 0. This means we look at values of x that are slightly less than 0. The slope (derivative) for these values is -1.
Step 5: Calculate the right-hand derivative at x = 0. This means we look at values of x that are slightly greater than 0. The slope (derivative) for these values is 1.
Step 6: Compare the left-hand and right-hand derivatives. The left-hand derivative is -1 and the right-hand derivative is 1.
Step 7: Conclude that since the left-hand and right-hand derivatives are not equal, the derivative of f(x) = |x| at x = 0 is undefined.
