The Power Rule (y = x^n)
We've worked out four derivatives one at a time, each from the difference quotient. Lined
up, they're hiding a pattern:
\frac{d}{dx}[1] = 0 \qquad \frac{d}{dx}[x] = 1 \qquad \frac{d}{dx}[x^2] = 2x \qquad \frac{d}{dx}[x^3] = 3x^2
Look at what happens to the exponent. Reading
x^0,
x^1,
x^2,
x^3,
each derivative brings the old power down to the front as a multiplier and
drops the power by one.
The pattern, stated once
That's the power rule. For any power n:
\frac{d}{dx}\bigl[\,x^n\,\bigr] = n\,x^{\,n-1}
Multiply by the exponent, then subtract one from the exponent. Check it against the list:
x^3 \to 3x^2 (✓), x^2 \to 2x^1 = 2x
(✓), x^1 \to 1\cdot x^0 = 1 (✓), and even
x^0 \to 0\cdot x^{-1} = 0 (✓) — the constant rule falls right out.
Try every power at once
Slide n and watch the curve x^n (solid)
next to its derivative n\,x^{n-1} (dashed). The derivative is
always one degree lower — and notice the dashed curve is exactly the slope of the solid one
at every point.
The derivative is the slope
Here's y = x^3 (solid) with the straight tangent line that just
grazes it (dashed). Slide the point a along the curve: the power
rule says the slope there is 3a^2, and the tangent's steepness
matches it everywhere.
It isn't just for whole numbers
The rule keeps working when n is negative or a
fraction — you just have to rewrite the function as a power first. A
reciprocal and a square root are both secretly powers:
\frac{1}{x} = x^{-1} \;\Rightarrow\; \frac{d}{dx}\!\left[\frac{1}{x}\right] = -1\,x^{-2} = -\frac{1}{x^2}
\sqrt{x} = x^{1/2} \;\Rightarrow\; \frac{d}{dx}\bigl[\sqrt{x}\bigr] = \tfrac{1}{2}\,x^{-1/2} = \frac{1}{2\sqrt{x}}
Same recipe every time: power to the front, drop it by one.