Check whether the function is defined at the point of interest
Verify that the function is continuous at the point
Compute the derivative using the limit definition
Confirm that the one-sided derivatives exist and are equal
Check for corners, cusps, vertical tangents, or discontinuities
For piecewise functions, test differentiability at the boundary points
Ensure the derivative limit is finite
Compare left-hand and right-hand slopes at the point
Use known differentiability rules for sums, products, quotients, and compositions
Confirm that the function is differentiable on an interval by checking every point in the interval