Derivative of a Power Function
One of the most fundamental and frequently used rules in all of calculus — the Power Rule — gives us an elegant shortcut for finding the derivative of any power function.
Why Do We Need This?
The derivative tells us the instantaneous rate of change of a function — the slope of the tangent line at any point. Power functions like $x^2$, $x^3$, $\sqrt{x}$, and $\frac{1}{x}$ appear everywhere in physics, engineering, and economics. The Power Rule lets us differentiate all of them instantly.
📐 Formal Definition: The Power Rule
A power function has the form:
$f(x) = x^n$
where $n$ is any real number. The Power Rule states:
$$\frac{d}{dx}\left[x^n\right] = n \cdot x^{n-1}$$
In words: "Bring the exponent down as a coefficient, then reduce the exponent by one."
More generally, with a constant multiplier (using the Constant Multiple Rule):
$$\frac{d}{dx}\left[c \cdot x^n\right] = c \cdot n \cdot x^{n-1}$$