Applying the Power Rule

The power rule works because the exponent counts how many times x multiplies itself, and differentiation measures how fast that product grows as x nudges forward. You can see this directly from the limit definition: expand (x+h)³ as x³ + 3x^2h + 3xh² + h³, subtract x³, divide by h, and let h shrink to zero. The x³ cancels, the h² and h³ terms vanish, and only 3x² survives. That leftover coefficient is exactly the original exponent, and the power it keeps is exactly one less than before. The same bookkeeping happens for any exponent, including negative and fractional ones, which is why d/dx[xⁿ] = n x^n-1 holds universally.

The real work in this topic is rewriting everything as x to a power before you touch the rule. Square roots become x^1/2, cube roots become x^1/3, and anything in a denominator becomes a negative exponent: 1/x³ is x^−3. Once every term looks like a coefficient times xⁿ, you differentiate term by term, since the derivative of a sum is the sum of the derivatives. Each term's exponent slides down front as a new coefficient, multiplied by whatever constant was already there, and the exponent itself drops by one.

This topic is the engine behind every derivative computation that follows: product rule, quotient rule, and chain rule all eventually reduce to differentiating powers of x. Getting comfortable rewriting radicals and fractions as exponents now pays off immediately, since later rules combine with the power rule constantly rather than replacing it.