Educational tool; results are estimates for learning, verify with your textbook or instructor.
This tool estimates the slope of a simple function at one point by a difference quotient. You type an expression in $x$, a point $x_0$, and a step size $h$. The number you get is a numerical estimate of $f'(x_0)$ when $f$ is smooth enough near that point.
The estimate is meaningful when $f$ is defined at the sample points the method uses, and when $h$ is small enough that the secant looks like the tangent, yet large enough that rounding does not swallow the change in $f$.
This is not a computer algebra system. It does not simplify a formula, it does not prove that $f$ is differentiable, and it does not solve an implicit equation $F(x,y)=0$. For the meaning of the number itself, read what a derivative actually measures.
Allowed pieces: numbers, $x$, plus, minus, times, divide, powers, parentheses, and $\sin$, $\cos$, $\tan$, $\mathrm{sqrt}$, $\mathrm{abs}$, $\log$ (natural), $\exp$, $\pi$, $e$. Type them as sin, cos, tan, sqrt, abs, log, exp, pi, e.
How to use the estimator
Start with a function you already know. The default $x^2$ at $x_0=3$ with $h=0.001$ is a good first check. The true derivative is $2x$, so a trustworthy run should land near $6$. If it does not, check the expression spelling before you trust a harder example.
Pick a method. Central difference samples both sides and is usually the most accurate on a smooth graph. Forward difference only looks to the right of $x_0$. Backward difference only looks to the left. Use a one sided method when the formula is missing on the other side, such as $\sqrt{x}$ at $0$.
Press Estimate slope. The large number is the estimate. The note under it repeats the difference formula and the sampled values, so you can see what was subtracted.
How to read the output
Treat the output as a probe, not a certificate. Cut $h$ by a factor of ten and run again. If the estimate barely moves, you are probably near a stable slope. If it jumps around as $h$ shrinks, you are seeing rounding noise or a corner the tangent cannot settle on.
Units still belong to you. If $f$ is position in meters and $x$ is time in seconds, the estimate is meters per second. The page does not know units. You attach them after you know what $f$ measures.
Common pitfalls
A huge $h$ measures a long secant, not a tangent. A tiny $h$, such as $10^{-16}$, can make $x_0$ and $x_0+h$ identical in floating point, so the quotient becomes junk. Stay in a band like $10^{-2}$ down to $10^{-6}$ unless you have a reason to leave it.
A jump, a corner, or a vertical tangent will not settle no matter how carefully you pick $h$. The parser only accepts the short list above. It is not general code, and it will reject extra names.
A central difference at a point that is defined on only one side can look worse than a one sided difference. Match the method to the domain you actually have.
Related reading
Read what a derivative actually measures before you treat the output as a rate. If the slope is hidden in an equation rather than an explicit $y=f(x)$, use implicit differentiation intuition. For the limit idea behind the difference quotient, see epsilon delta intuition and the derivative glossary entry.
The matching map is calculus foundations. A course shaped path is the AP Calculus AB bridge. Nearby tools: numerical limit estimator and Riemann sum calculator.
