Derivative at a point
Compute the derivative of a function at a point and see the tangent line – its slope is the derivative.
Supported: + − * / ^, functions sin, cos, tan, exp, ln, log, sqrt, abs. Constants pi, e. Multiplication can be written as 2x.
What a derivative is
The derivative of a function at a point gives the instantaneous rate of change of the function – how steeply it rises or falls at that spot. Geometrically it's the slope of the tangent line to the graph at that point. When the derivative is positive, the function is increasing; when negative, decreasing; when zero, the function has a peak, trough, or saddle at that point.
f′(x₀) = limh→0 [ f(x₀+h) − f(x₀) ] / h
Practical uses of the derivative
- Physics: velocity is the derivative of position with respect to time, acceleration is the derivative of velocity.
- Finding maxima and minima – where the derivative is zero (optimising cost, profit, shape).
- Economics: marginal cost and marginal utility are derivatives of the total quantities.
- Machine learning: the gradient (derivative) tells you which direction to adjust the model to improve it.
Tangent to the graph
The tangent is a line that touches the graph at one point and has the same slope as the function there. Its equation is y = f(x₀) + f′(x₀)·(x − x₀). On the graph above you see the function in green and the tangent in orange – their slope at the point of tangency is the same, which is exactly the value of the derivative.
Tip: for polynomials and basic functions we compute the derivative exactly (symbolically), for more complex ones numerically. Symbolic derivative shows the derivative formula.