Derivative rules

Constant rule \frac{d}{dx}\, c = 0 The constant rule says the derivative of any constant is 0. The formula, a one-line proof, examples with π and e, and the mistakes to avoid. Power rule \frac{d}{dx}\, x^n = n\,x^{n-1} The power rule: the derivative of x^n is n·x^(n−1). Formula, examples with negative and fractional powers, a short proof and common mistakes. Constant multiple rule \frac{d}{dx}\,[c\,f(x)] = c\,f'(x) The constant multiple rule: the derivative of c·f(x) is c·f'(x). The formula, worked examples, a quick proof and the mistakes students make with it. Sum and difference rule \frac{d}{dx}\,[f(x) \pm g(x)] = f'(x) \pm g'(x) The sum and difference rule: differentiate a sum or difference one term at a time. The formula, worked examples, a proof and where students go wrong. Product rule \frac{d}{dx}\,[f(x)\,g(x)] = f'(x)\,g(x) + f(x)\,g'(x) The product rule: the derivative of f·g is f'g + fg'. The formula, worked examples, the three-factor version, a short proof and common mistakes. Quotient rule \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2} The quotient rule: (f/g)' = (f'g − fg')/g². The formula, the low d-high mnemonic, worked examples, when to avoid it and common mistakes. Chain rule \frac{d}{dx}\,f(g(x)) = f'(g(x))\,g'(x) The chain rule: the derivative of f(g(x)) is f'(g(x))·g'(x). The formula in both notations, a table of common forms, worked examples and mistakes. Implicit differentiation \frac{d}{dx}\,f(y) = f'(y)\,\frac{dy}{dx} Implicit differentiation finds dy/dx when y is not solved for. The key rule, a four-step method, worked examples with tangent lines and common mistakes. Logarithmic differentiation \frac{dy}{dx} = y \cdot \frac{d}{dx}\big[\ln y\big] Logarithmic differentiation: take ln of both sides, differentiate, multiply by y. Steps and examples for x^x, variable powers and messy products. Inverse function rule (f^{-1})'(x) = \frac{1}{f'\big(f^{-1}(x)\big)} The inverse function rule: the derivative of f⁻¹ at x is 1/f'(f⁻¹(x)). The formula, why slopes are reciprocals, worked examples and arcsin and ln.