SL 5.3—Derivative of powers

The power rule differentiates xⁿ by lowering the exponent.

For a constant n, d(xⁿ)/dx=nxⁿ⁻¹; the factor n records how rapidly the power changes.

Example

d(3x⁵−2x²)/dx=15x⁴−4x.

Apply $d(ax^n)/dx=anx^{n-1}$ term by term for integer exponents in this objective, retaining coefficients and differentiating constants to zero.

The derivative of $x^0$ is zero. Rational powers and chain-rule factors belong to SL 5.6, so do not import them into this integer-power step.