ddxxn=nxn-1 for rational n where the derivative exists. The same power rule applies after algebraic rewriting, so roots and reciprocals need no separate derivative rule. Method Rewrite first, multiply by the exponent, reduce the exponent by one, simplify, and keep the original restrictions. Example y= 3sqrt(x)=3x-1/2, dydx=- 32x3/2 (x>0) Exam check A constant differentiates to zero. A root may exist at an endpoint even when its derivative there is undefined.

\(\dfrac{\mathrm{d}}{\mathrm{d}x}x^n=nx^{n-1}\) for rational \(n\) where the derivative exists.

The same power rule applies after algebraic rewriting, so roots and reciprocals need no separate derivative rule.

Rewrite first, multiply by the exponent, reduce the exponent by one, simplify, and keep the original restrictions.

\(y=\dfrac{3}{\sqrt{x}}=3x^{\frac{-1}{2}},\quad\dfrac{\mathrm{d}y}{\mathrm{d}x}=-\dfrac{3}{2x^{\frac{3}{2}}}\quad(x>0)\)

A constant differentiates to zero. A root may exist at an endpoint even when its derivative there is undefined.

Roots and reciprocals as powers

Roots and reciprocals as powers