Back to Calculus
Calculus › Derivatives
Derivatives
The derivative of a function tells you the slope of its graph at any point — the rate of change.
Slope at a point
f(x) = ¼x²
x position2.0
f(x) = ¼x² · point
(2.0, 1.00)
f′(x) = ½x · slope here
1.00
Slope, but for curves
A straight line has a single slope. A curve has a different slope at every point. The derivative f′(x) gives you that slope as a function of x.
Some quick rules
(xⁿ)′ = n · xⁿ⁻¹(the power rule).- The derivative of a constant is 0.
- Derivatives add:
(f + g)′ = f′ + g′. - Constants pull out:
(c · f)′ = c · f′.
So for f(x) = ¼x²: f′(x) = ½x. At x = 4 the slope is 2. The slider shows that.
Why slopes matter
Velocity is the derivative of position. Acceleration is the derivative of velocity. Marginal cost is the derivative of cost. Slope shows up everywhere in physics, economics, and engineering.
Maxima and minima
The derivative is zero where a smooth curve hits a peak or a valley. That's how calculus finds extremes — by solving f′(x) = 0 and checking which solutions are tops vs bottoms.