Higher-Order Derivative
A higher-order derivative is the result of differentiating a function more than once, such as the second or third derivative.
Formula
f''(x) = \frac{d^2y}{dx^2}
Definition
A higher-order derivative means differentiating a function more than once: the second derivative is the derivative of the derivative, and so on. The $n$th derivative $f^{(n)}(x)$ is defined inductively ($f^{(0)} = f$, $f^{(n)} = (f^{(n-1)})'$) and is obtained by differentiating $f$ repeatedly $n$ times, written $f''$, $f'''$, $f^{(4)}, \ldots$ or $d^ny/dx^n$. A function is $C^n$ if its first $n$ derivatives all exist and are continuous, and $C^\infty$ (smooth) if all derivatives exist.
Example
If position is $s(t) = t^3$, velocity $s'(t) = 3t^2$ and acceleration $s''(t) = 6t$ are the first and second derivatives. For $f(x) = x^4$: $f' = 4x^3$, $f'' = 12x^2$, $f''' = 24x$, $f^{(4)} = 24$, $f^{(5)} = 0$, showing that polynomials eventually differentiate to zero. In contrast, for $f(x) = e^x$, $f^{(n)}(x) = e^x$ for all $n$, which is why $e^x$ equals its own Taylor series everywhere.
Key Insight
Each time you differentiate, you learn something new: first derivative is rate of change, second derivative is rate of the rate of change, and the second derivative also reveals concavity. The coefficients of a Taylor series are determined by higher-order derivatives at a point, $f^{(n)}(a)/n!$, and in PDEs the order of the highest derivative present determines the equation's type (elliptic, parabolic, hyperbolic) and the nature of its solutions.