Derivative Calculator

It computes the numeric value of the first derivative f'(x) of a single-variable function at one specific point, not a symbolic formula. You enter a function such as x^2, sin(x), or 3*x^3 - 2*x + 1 and the point where you want the slope. The tool returns f'(x), the function value f(x), and the equation of the tangent line at that point.

The central-difference (symmetric difference quotient) method estimates the slope with f'(a) = (f(a + h) - f(a - h)) / (2h), using a tiny step h on each side of the point. Sampling both sides cancels the first-order error, so its accuracy is order h squared, which is much better than a one-sided difference. The calculator picks h automatically based on the magnitude of your point to balance rounding and truncation error.

Use x as the variable and standard math notation: + - * / for arithmetic, ^ for powers, and parentheses for grouping. Supported functions include sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, sqrt, cbrt, ln (natural log), log (base 10), exp, and abs. The constants pi and e are available too, so an entry like 2*sin(pi*x) works.

The tangent line is the straight line that just touches the curve at the chosen point, and its slope is exactly the derivative f'(x) at that point. The calculator reports the slope and the full line in the form y = m*x + b, where m equals the derivative and b is the y-intercept. This is the line you would draw to show the instantaneous rate of change.

The calculator returns no result when the function is not defined at or right around your point, or when the slope is not a finite number. For example, ln(x) at x = 0 or 1/x at x = 0 are undefined, so no derivative is reported. Check that your expression is well formed and that the point lies inside the function's domain.

For smooth functions the central-difference estimate typically matches the exact derivative to several decimal places, and it is exact for straight lines and quadratics. Because it is a numeric approximation rather than symbolic differentiation, very small deviations can appear in the last digits. For polynomials, exponentials, logarithms, and trig functions, the displayed value is reliable for everyday use.