Powers, roots and polynomials

Derivative of x \frac{d}{dx}\,x = 1 The derivative of x is 1. See why with the limit definition and the power rule, plus derivatives of 5x, ax + b and x with respect to other variables. Derivative of x² \frac{d}{dx}\,x^2 = 2x The derivative of x² is 2x. See the limit definition proof, the chain rule version for u², a graph, tangent lines and worked examples like (3x + 1)². Derivative of √x \frac{d}{dx}\,\sqrt{x} = \frac{1}{2\sqrt{x}} The derivative of √x is 1/(2√x). See the power rule and limit proofs, the chain rule for √u, why x = 0 fails, and examples like √(x² + 1). Derivative of ∛x \frac{d}{dx}\,\sqrt[3]{x} = \frac{1}{3\sqrt[3]{x^2}} The derivative of the cube root of x is 1/(3∛(x²)). See the power rule and limit proofs, the vertical tangent at 0 and chain rule examples. Derivative of 1/x \frac{d}{dx}\,\frac{1}{x} = -\frac{1}{x^2} The derivative of 1/x is -1/x². See the limit definition proof, the power rule with x⁻¹, the chain rule for 1/u and worked examples like 1/(2x + 5). Derivative of 1/x² \frac{d}{dx}\,\frac{1}{x^2} = -\frac{2}{x^3} The derivative of 1/x² is -2/x³. See the power rule and quotient rule proofs, the chain rule for 1/u², a graph and examples like 1/(3x - 2)². Derivative of |x| \frac{d}{dx}\,\lvert x\rvert = \frac{x}{\lvert x\rvert} The derivative of |x| is x/|x|: 1 for x > 0, -1 for x < 0, and undefined at 0. See the proof, the corner at 0 and the chain rule for |u|. Derivative of a constant \frac{d}{dx}\,c = 0 The derivative of a constant is 0. See why with the limit definition, plus tricky cases like π², e³, ln 5 and constants multiplied by functions.