When you use a derivative calculator like ours, it isn't doing the algebra you learned in class. It's doing numerical differentiation: estimating the slope of a curve at one point by sampling the function nearby. Understanding that is the difference between trusting your answer and being fooled by it.
On this page
What d/dx returns
The syntax is d/dx(f(x), a). It returns one number: the instantaneous rate of change of f at x = a — geometrically, the slope of the tangent line to the curve at that point. It does not return a derivative function like 3x². If you want the slope at three different points, you run it three times.
How it works: the central difference
The derivative is defined as a limit of the difference quotient. A numerical engine can't take a true limit, so it picks a very small step h and uses the symmetric (central) difference:
f'(a) ≈ [ f(a + h) − f(a − h) ] / (2h)Sampling on both sides of a (rather than just one) cancels the leading error term, so for smooth functions the estimate is accurate to many decimal places. This is why a well-behaved case like the derivative of x³ comes back essentially exact.
Worked examples
d/dx(x^3, 2) → 12. Symbolically (x³)′ = 3x², and 3·2² = 12. ✔
d/dx(x^2, 5) → 10. (x²)′ = 2x, so 2·5 = 10.
d/dx(sin(x), 0) → 1. (sin x)′ = cos x, cos 0 = 1.
d/dx(e^(x), 0) → 1. (eˣ)′ = eˣ, and e⁰ = 1.
d/dx(ln(x), 1) → 1. (ln x)′ = 1/x, 1/1 = 1.
Numerical vs symbolic
| Symbolic | Numerical (this tool) | |
|---|---|---|
| Returns | A formula, e.g. 3x² | A number at one point, e.g. 12 |
| Exactness | Exact | Estimate (usually many correct digits) |
| Needs a point? | No | Yes — you supply x = a |
| Speed | Can be slow / complex | Instant |
For most homework checking and engineering work, the numerical value is exactly what you want. To see how the same engine handles integration, read the pillar guide on integration & derivatives.
Two pitfalls
At a non-differentiable point the answer is meaningless. For |x| at x = 0 the true derivative doesn't exist, yet the symmetric formula returns 0 because the left and right samples are equal. Always confirm the function is smooth at the point.
(sin x)′ = cos x only in radians. In Deg mode the derivative is scaled by π/180 and your answer will be wrong. Switch to Rad first.
A result like 11.99999999 is the numerical estimate of 12 — round it. This is expected behaviour, not an error.
Frequently asked questions
What does d/dx(f(x), a) return?
The slope of f at the single point x = a. For example d/dx(x^3, 2) returns 12.
How does numerical differentiation work?
Via a central difference: f'(a) ≈ [f(a+h) − f(a−h)] / (2h) for a very small h.
Why is my derivative slightly off, like 2.0000001?
It's a numerical estimate, not a symbolic result. Tiny end-digit rounding is normal — round sensibly.
Can it differentiate a function with a corner?
Be careful: at points like |x| at x = 0 the derivative doesn't exist, but the formula may still return 0.
See it for yourself
Open the calculator, switch to Rad mode, and try d/dx(sin(x), 0).