Check whether the function is continuous at the point
Compute the derivative using the limit definition
Verify that the limit of the difference quotient exists
Check that the left-hand and right-hand derivatives are equal
Ensure the function has no corner, cusp, jump, or vertical tangent at the point
Confirm the function is defined in a neighborhood of the point
For piecewise functions, check differentiability at the boundary points
If the function is built from differentiable functions, use differentiation rules to confirm differentiability
If the function is smooth, it is differentiable
If the derivative exists at every point in an interval, the function is differentiable on that interval