Derivative Calculator

The derivative of a function tells you its instantaneous rate of change at a specific point. If you plot the function on a graph, the first derivative gives the slope of the tangent line at that point.

Central Difference Method:

Instead of using the limit definition directly, this calculator approximates derivatives with finite differences:

• First derivative: f'(x) ≈ [f(x + h) - f(x - h)] / (2h)
• Second derivative: f''(x) ≈ [f(x + h) - 2f(x) + f(x - h)] / h²

The central difference approach is more accurate than one-sided differences because the errors from each side partially cancel out. With the default step size of h = 0.0001, results are typically accurate to 8 or more significant figures.

Understanding First vs. Second Derivatives:

• f'(x) > 0 means the function is increasing at that point
• f'(x) < 0 means the function is decreasing
• f'(x) = 0 suggests a possible local maximum, minimum, or inflection point
• f''(x) > 0 means the curve is concave up (like a bowl), so a critical point here is a local minimum
• f''(x) < 0 means the curve is concave down (like a hill), so a critical point here is a local maximum

Practical Examples:

If f(x) = x^2, then f'(3) = 6 (the slope at x = 3 is 6) and f''(3) = 2 (constant positive concavity, confirming the parabola opens upward). For f(x) = sin(x), f'(0) = 1 and f''(0) = 0, meaning the sine curve is rising with a slope of 1 at the origin and is at an inflection point.