Prove the identity for $x0$, where every real rational power is defined and differentiabl

Question

Prove the identity for $x>0$, where every real rational power is defined and differentiable. Extensions to negative inputs or zero require separate domain and differentiability checks. Let ${\mathrm{f}(x) = x^{n}}$, where ${n}$ is a rational number. Show that ${\mathrm{f}'(x) = nx^{n - 1}}$ by using differentiation from first principles.

Accepted Answer

$f'(x)=nx^{n-1}$ for rational $n$ and $x>0$, proved by the difference quotient, including positive, zero and negative numerators.

Techniques of Differentiation - Differentiation by First Principles

Techniques of Differentiation - Differentiation by First Principles

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