Result
| Step h | x - h | f(x - h) | x + h | f(x + h) | Left Derivative | Right Derivative | Symmetric Derivative |
|---|
Calculator
Use abs(x) for |x|, x^2 for powers, and standard functions like sin(x), cos(x), exp(x), sqrt(x), and log(x).
Example Data Table
Example using f(x) = x2 at a = 2. The derivative should approach 4.
| h | x - h | f(x - h) | x + h | f(x + h) | Left Derivative | Right Derivative | Symmetric Derivative |
|---|---|---|---|---|---|---|---|
| 0.5 | 1.5 | 2.25 | 2.5 | 6.25 | 3.5 | 4.5 | 4 |
| 0.25 | 1.75 | 3.0625 | 2.25 | 5.0625 | 3.75 | 4.25 | 4 |
| 0.125 | 1.875 | 3.515625 | 2.125 | 4.515625 | 3.875 | 4.125 | 4 |
Formula Used
Continuity test at x = a:
lim(x→a-) f(x) = f(a) = lim(x→a+) f(x)
Left derivative estimate:
[f(a) - f(a - h)] / h
Right derivative estimate:
[f(a + h) - f(a)] / h
Symmetric derivative estimate:
[f(a + h) - f(a - h)] / (2h)
Differentiability rule:
A function is differentiable at a point when it is continuous there and the left and right derivatives match.
How to Use This Calculator
- Enter the function in terms of x.
- Enter the point where you want to test differentiability.
- Choose an initial step size h.
- Set how many shrinking step levels you want.
- Enter a tolerance for comparison.
- Use the optional f(a) override when the function value is defined separately.
- Press Calculate to show the result above the form.
- Review the table, then download CSV or PDF if needed.
About Differentiability at a Point
What It Means
Differentiability at a point tells you whether a function has one clear slope at a chosen x-value. It is a central idea in calculus. A function may be continuous and still fail to be differentiable. Sharp corners, cusps, vertical tangents, and breaks can all cause trouble.
This differentiability at a point calculator helps you examine left-hand derivative, right-hand derivative, continuity, and symmetric slope estimates. You enter a function, choose the target point, set a step size, and compare nearby function values. The tool then checks whether both one-sided slopes move toward the same number. That matching behavior suggests differentiability.
Why It Matters
Students use point differentiability in limit questions, graph analysis, and tangent line problems. Teachers use it to explain why smooth graphs behave differently from sharp graphs. It is also useful in numerical work, because local smoothness improves approximation and prediction near the point being studied.
A differentiable function near a point often behaves in a stable way. Small changes in x lead to manageable changes in output and slope. That makes local modeling easier. The idea appears in physics, economics, engineering, and many applied maths settings.
How This Calculator Helps
The calculator first estimates f(a), the function value at the chosen point. Next, it samples points slightly left and right of a. It computes left and right difference quotients. Then it checks continuity and compares both one-sided derivative estimates against your selected tolerance.
If continuity fails, differentiability fails immediately. If the one-sided derivatives disagree, the function is not differentiable at that point. If both estimates match closely, the tool reports likely differentiability. Because this is a numerical test, smaller step sizes usually give better insight.
Best Practice
Use this page for homework, revision, and concept checks. It works well for polynomial, trigonometric, exponential, logarithmic, and absolute value expressions. You can also study removable issues by entering an override for the actual point value. Watch the result table carefully. Stable estimates support smoothness. Unstable estimates often reveal a corner, cusp, or discontinuity.
FAQs
1. What does differentiable at a point mean?
It means the function has one well-defined instantaneous rate of change at that exact point. Numerically, the left and right derivative estimates should approach the same value.
2. Can a function be continuous but not differentiable?
Yes. A function can be continuous and still fail the derivative test. Absolute value at zero is the classic example because the graph has a sharp corner.
3. Why does the calculator test continuity first?
If a function is not continuous at the point, it cannot be differentiable there. Continuity is a necessary condition before comparing one-sided derivatives.
4. What is the purpose of the optional f(a) override?
It lets you define the actual point value separately. This is useful for piecewise functions or removable cases where the expression alone does not supply the intended value at the point.
5. What does the symmetric derivative show?
It gives another numerical slope estimate using values on both sides of the point. It is helpful for comparison, but the left and right checks remain the main differentiability test.
6. Why do smaller step sizes matter?
Smaller h values usually move the estimate closer to the true local behavior. They help you see whether the derivative values stabilize as the sampled points approach the target.
7. Can this calculator prove differentiability exactly?
No. It gives a strong numerical indication. Exact proof may still require algebra, limits, or piecewise analysis, especially for difficult functions.
8. Which expressions work well here?
It works best with common real-valued expressions such as polynomials, trig functions, exponentials, logs, roots, and absolute value written in standard calculator syntax.