SL 5.3—Derivative of powers

For $f(x)=ax^n$ with integer $n$, $f'(x)=anx^{n-1}$. Differentiate a sum term by term and treat a constant as having derivative zero.

Negative integer powers are included where the original function is defined: $d(x^{-2})/dx=-2x^{-3}$. Keep coefficients, signs and domain restrictions visible before simplifying.

Example

For $f(x)=4x^3-2x^{-1}+7$, $f'(x)=12x^2+2x^{-2}=12x^2+2/x^2$, with $x\ne0$ inherited from the original function.

This SL rule is bounded to integer exponents. Rational powers, chain, product and quotient rules belong to AHL 5.9; do not import them into this Objective.