Overview
The derivative measures a function's instantaneous rate of change at a point and equals the slope of its tangent line there. It appears in velocity and acceleration analysis, optimization, and the study of increasing and decreasing behavior. This calculator estimates the derivative numerically from nearby function values, then shows the slope and the tangent and normal line equations.
Formula
f'(a) = lim(h→0) [f(a + h) − f(a − h)] ÷ 2h, and the tangent is y = f'(a)(x − a) + f(a).
How it works
Write the function
Use x and supported functions such as sin, ln and sqrt.
Choose the point
Enter the value a where the derivative is needed.
Estimate the instantaneous slope
The calculator applies a central difference to obtain the first derivative.
Read the interpretation
The second derivative, function behavior, tangent and normal lines are shown.
Worked examples
Example 1: f(x) = x² + 3x at x = 2
f'(x) = 2x + 3, so f'(2) = 7, and the tangent is y = 7x − 4.
Example 2: f(x) = sin x at x = 0
f'(0) = cos 0 = 1, and the tangent is y = x.
Frequently asked questions
Is the result a symbolic or numeric derivative?
The calculator returns a numerical derivative at the selected point, not the symbolic derivative expression.
Why is the second derivative shown?
It describes concavity and helps distinguish local maxima from minima.
What happens at a sharp corner?
The left and right derivatives may differ, so the derivative does not exist at that point.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.