## University Calculus: Early Transcendentals (3rd Edition)

We can rewrite any real power of x as a power of $e$, like so: $x^n= e^{n\ln x}$ with the restriction $x\gt0$ On taking the derivative, we get: $\frac{d(x^n)}{dx}=\frac{d(e^{n\ln x})}{dx}$ Applying the chain rule, gives us: $\frac{d(x^n)}{dx}=(e^{n\ln x})\frac{d({n\ln x})}{dx}$ $\frac{d(x^n)}{dx}=(e^{n\ln x})\frac{({n})}{x}$ $\frac{d(x^n)}{dx}=(x^n)\frac{({n})}{x}$ $\frac{d(x^n)}{dx}=nx^{(n-1)}$ Which is the power rule for derivatives.