Why the Power Rule Turns x^n into n·x^(n−1)

The power rule is one of the main shortcuts in differential calculus. For a power function

$$f(x)=x^n,$$

its derivative is

$$f'(x)=nx^{n-1}.$$

The exponent becomes a coefficient, and the exponent on $x$ decreases by one.

The pattern comes from the difference quotient

The derivative starts from

$$\frac{(x+h)^n-x^n}{h}.$$

When $(x+h)^n$ is expanded, the $x^n$ terms cancel. Every remaining term contains at least one factor of $h$. After dividing by $h$, the only term that remains nonzero as $h\to0$ is

$$nx^{n-1}.$$

That is the structural reason the coefficient $n$ appears and the exponent drops by one.

Use the rule term by term in polynomials

For example, if

$$p(x)=4x^5-2x^2+7,$$

then

$$p'(x)=20x^4-4x.$$

The constant differentiates to zero, and each power term follows the same rule.

The exponent rule extends beyond positive integers

Calculus extends the same pattern to many negative and fractional powers, provided the function is considered on a domain where those powers and derivatives are defined. Domain restrictions therefore matter more once reciprocals or roots appear.