Find the derivative of f(x) = x^2 sin(1/x) at x = 0.

Find the derivative of f(x) = x^2 sin(1/x) at x = 0.

Step 1: Understand that we want to find the derivative of the function f(x) = x^2 sin(1/x) at the point x = 0.
Step 2: Recall the limit definition of the derivative: f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h.
Step 3: In our case, we need to find f'(0), so we set a = 0. We need to calculate f(0 + h) - f(0).
Step 4: Calculate f(0). Since f(x) = x^2 sin(1/x) is not defined at x = 0, we define f(0) = 0 to make it continuous.
Step 5: Now, we need to find f(h) = h^2 sin(1/h) for h ≠ 0.
Step 6: Substitute into the limit: f'(0) = lim (h -> 0) [h^2 sin(1/h) - 0] / h.
Step 7: Simplify the expression: f'(0) = lim (h -> 0) h sin(1/h).
Step 8: As h approaches 0, sin(1/h) oscillates between -1 and 1, so h sin(1/h) approaches 0.
Step 9: Therefore, we find that f'(0) = 0.
Step 10: Since the derivative exists and is equal to 0, we conclude that f(x) is differentiable at x = 0.

Limit Definition of Derivative – The derivative at a point is defined as the limit of the difference quotient as the interval approaches zero.
Piecewise Functions – Understanding how to handle functions that behave differently at certain points, such as f(x) = x^2 sin(1/x) at x = 0.
Continuity and Differentiability – A function must be continuous at a point to be differentiable there, and the behavior of the function around that point is crucial.