Rules of calculus

Limit laws \lim_{x \to a}\,[f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) The limit laws let you split a limit into smaller limits. Every law with its conditions, worked examples and the mistakes that break them. Direct substitution \lim_{x \to a} f(x) = f(a) Direct substitution evaluates a limit by plugging in the value. Learn when it works, how to read the result and what to do with 0/0 or a/0. L'Hôpital's rule \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} L'Hôpital's rule evaluates 0/0 and ∞/∞ limits by differentiating the top and bottom. The exact conditions, worked examples and other indeterminate forms.

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.

Reverse power rule \int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \ne -1 The reverse power rule: the integral of x^n is x^(n+1)/(n+1) + C for n ≠ −1. Formula, examples with roots and negative powers, and the ln|x| exception. Linearity of the integral \int \left[a\,f(x) + b\,g(x)\right]dx = a\int f(x)\,dx + b\int g(x)\,dx The integral linearity rules: integrate sums term by term and pull constants out. Formulas, examples and the mistakes to avoid with products and quotients. U-substitution \int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x) The u-substitution rule reverses the chain rule: ∫f(g(x))g′(x)dx = ∫f(u)du. Formula, how to choose u, definite integral limits and examples. Integration by parts \int u\,dv = uv - \int v\,du The integration by parts formula ∫u dv = uv − ∫v du, how LIATE picks u, the definite integral version and short examples with ln x, x·eˣ and x·cos x. Definite integral properties \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx The properties of definite integrals: reversing limits, zero width, additivity over intervals, odd and even symmetry and comparison bounds, with examples.

Geometric series formula \sum_{n=0}^{\infty} a\,r^n = \frac{a}{1 - r}, \quad \lvert r\rvert < 1 The geometric series formula: a + ar + ar² + ... = a/(1 − r) when |r| < 1. Partial sums, the convergence condition and worked examples. nth-term test \lim_{n\to\infty} a_n \ne 0 \;\Longrightarrow\; \sum a_n \text{ diverges} The nth-term test: if the terms of a series do not approach 0, the series diverges. The statement, why it cannot prove convergence, and examples. Ratio test L = \lim_{n\to\infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert The ratio test: compute L = lim |a(n+1)/a(n)|. If L < 1 the series converges absolutely, if L > 1 it diverges, and L = 1 is inconclusive. Alternating series test b_n \ge b_{n+1} > 0,\ \lim_{n\to\infty} b_n = 0 \;\Longrightarrow\; \sum (-1)^{n+1} b_n \text{ converges} The alternating series test (Leibniz test): if bₙ decreases to 0, then the series of (−1)ⁿbₙ converges. Conditions, the error bound and examples.