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If your car speedometer reads 60 mph, that tells you your exact speed right now — but when you step on the gas pedal, how quickly is that speed itself climbing?
In calculus, the first derivative gives the rate of change of a curve, while the second derivative measures how fast that slope itself is changing.
How do we write this mathematically, and what does the notation actually represent?

Notation and Concept

In the 1670s, mathematician Gottfried Wilhelm Leibniz introduced the symbol d2ydx2\frac{d^2y}{dx^2} as shorthand for applying the derivative operator twice: ddx(dydx)\frac{d}{dx}\left(\frac{dy}{dx}\right).
In function notation, we write the second derivative as f′′(x)f''(x), read aloud as ff double-prime of xx.
How do we compute it with algebraic functions? Let's work through a standard polynomial.

Worked Example: Polynomials

Find d2ydx2\frac{d^2y}{dx^2} for the curve y=3x4−5x3+7x−2y = 3x^4 - 5x^3 + 7x - 2.
First, apply the power rule to find the first derivative: dydx=12x3−15x2+7\frac{dy}{dx} = 12x^3 - 15x^2 + 7.
Next, differentiate this result a second time with respect to xx to get d2ydx2=36x2−30x\frac{d^2y}{dx^2} = 36x^2 - 30x.
A common mistake is confusing d2ydx2\frac{d^2y}{dx^2} with (dydx)2\left(\frac{dy}{dx}\right)^2 — the second derivative is finding the derivative twice, not squaring the first derivative.