Calculated Result
| Metric | Value |
|---|
Calculator
Graph
Example Data Table
| Function | Point | Typical Result | Reason |
|---|---|---|---|
| abs(x) | 0 | Non differentiable | Corner formed by unequal one sided derivatives. |
| x^(2/3) | 0 | Non differentiable | Cusp with opposite infinite slope behavior. |
| x^(1/3) | 0 | Non differentiable | Vertical tangent with unbounded matching slope direction. |
| 1/x | 0 | Non differentiable | Discontinuity and undefined function value. |
| x^2 | 1 | Differentiable | Continuous point with equal one sided derivatives. |
Formula Used
The calculator compares continuity and one sided derivative behavior near the selected point.
Left derivative: f′-(a) = limh→0+ [f(a) - f(a - h)] / h
Right derivative: f′+(a) = limh→0+ [f(a + h) - f(a)] / h
Continuity check: left limit = right limit = f(a)
A point is differentiable when the function is continuous and both one sided derivatives match within the selected tolerance.
It is non differentiable when the point shows a discontinuity, corner, cusp, or vertical tangent.
How to Use This Calculator
- Enter the function using x as the variable.
- Type the point where you want to test differentiability.
- Choose an initial step, tolerance, and graph range.
- Press Analyze Point to compute the result.
- Review continuity, one sided derivatives, and classification.
- Use the graph to visually confirm sharp turns or breaks.
- Download the summary as CSV or PDF when needed.
Understanding Non Differentiability
A non differentiability calculator helps you test whether a function fails to have a derivative at a selected point. In calculus, a derivative exists when the left hand and right hand rates of change agree. If they do not match, the graph has a special feature that deserves attention.
Why non differentiability happens
Non differentiability usually appears for four main reasons. A graph may have a corner, a cusp, a vertical tangent, or a discontinuity. Each case blocks the derivative in a different way. A corner happens when two slopes meet sharply. A cusp appears when the graph changes direction with extreme steepness. A vertical tangent occurs when the slope grows without bound. A discontinuity breaks the graph, so continuity fails before differentiability is even considered.
How this calculator works
This calculator estimates left hand and right hand derivatives numerically. It samples nearby points around the chosen value and compares the resulting secant slopes. It also checks continuity by comparing the function value with nearby left and right limits. When these values disagree, the tool reports likely non differentiability and suggests the most probable reason.
Why graphs matter
A graph makes the diagnosis easier. You can see whether the curve bends smoothly, forms a sharp point, rises vertically, or jumps. Numerical results are important, but visual confirmation often explains the behavior faster. This is helpful in homework, exam review, and concept practice.
Best uses for students
Use this tool for absolute value functions, radicals, piecewise expressions, and functions with unusual slope behavior. It is also useful when checking where a derivative formula may fail. Always enter the function carefully and choose a point close to the suspected feature.
Final thought
Differentiability is more than a rule. It is a test of local smoothness. When the left and right behavior disagree, the derivative tells you that the graph is not smooth there. This calculator turns that idea into a fast, visual, and practical workflow.
For reliable results, start with a moderate step size, then reduce it. Consistent classifications across smaller steps usually indicate the point has been identified correctly by the calculator.
FAQs
1) What does non differentiable mean?
A function is non differentiable at a point when its derivative does not exist there. This often happens at sharp turns, cusps, vertical tangents, or discontinuities.
2) Can a function be continuous but not differentiable?
Yes. Absolute value at zero is the standard example. The graph is continuous, but the left and right derivatives do not match.
3) Why are one sided derivatives important?
They show how the function changes from each side of the point. If the values disagree, the derivative at that point cannot exist.
4) How does the calculator detect a corner?
It estimates left and right derivatives numerically. When both are finite but unequal, the tool reports a likely corner or sharp turn.
5) What is the difference between a cusp and a vertical tangent?
A cusp usually shows opposite unbounded slope behavior from each side. A vertical tangent usually shows the same unbounded slope direction from both sides.
6) Can this tool test 1/x at zero?
Yes. The calculator will usually report a discontinuity or undefined point because the function value and nearby limits do not meet properly.
7) Why should I change the step size?
Numerical estimates depend on nearby sample spacing. Testing smaller steps helps confirm whether the classification remains stable near the selected point.
8) Is this calculator exact for every function?
No. It is a numerical aid. It gives strong practical evidence, but symbolic proof may still be needed in formal calculus work.