chain rule | integration by parts | l'Hôpital's rule | product rule | quotient rule | Simpson's rule | trapezoidal rule (total: 7)

The chain rule states that if a function g(x) is differentiable at the point x and a function f(x) is differentiable at the point g(x), then the composition f◦g is differentiable at x. Furthermore, let y = f(g(x)) and u = g(x), then dy/dx = dy/du·du/dx.

Integration by parts is a technique for performing indefinite integration integral udv or definite integration integral_a^budv by expanding the differential of a product of functions d(uv) and expressing the original integral in terms of a known integral integral vdu.

Suppose that lim(f(x)) and lim(g(x)) are both zero or both ±∞. Then l'Hôpital's rule states that if lim(f'(x)/(g'(x))) has a finite limit or the limit is ±∞, then lim(f(x)/(g(x))) = lim(f'(x)/(g'(x))).

The product rule states that for f and g two differentiable functions, d(f(x)g(x))/dx = f(x)g'(x) + g(x)f'(x), where f'(x) = df/dx denotes the derivative of f with respect to x.

The quotient rule states that for f and g two differentiable functions, d(f(x)/(g(x)))/dx = (g(x)f'(x) - f(x)g'(x))/g(x)^2, where f'(x) = df/dx denotes the derivative of f with respect to x.

Simpson's rule is a Newton-Cotes formula for approximating the integral of a function f using quadratic polynomials. Let f be tabulated at points x_0, x_1, and x_2 equally spaced by a distance h, and define f_n=f(x_n). Then Simpson's rule states that integral_(x_0)^(x_2)f(x)dx≈h(f_0 + 4f_1 + f_2)/3.

The trapezoidal rule is a Newton-Cotes formula for approximating the integral of a function f using linear segments. Let f be tabulated at points x_0 and x_1 spaced by a distance h, and write f_n=f(x_n). Then the trapezoidal rule states that integral_(x_0)^(x_1)f(x)dx≈h(f_0 + f_1)/2.

| formulation date | formulators | status chain rule | | | proved integration by parts | | | proved l'Hôpital's rule | | | proved product rule | | | proved quotient rule | | | proved Simpson's rule | 1615 (410 years ago) | Johannes Kepler | proved trapezoidal rule | | | proved | provers l'Hôpital's rule | Johann Bernoulli product rule | Gottfried Leibniz | additional people involved l'Hôpital's rule | Guillaume François Antoine Marquis de L'Hôpital Simpson's rule | Roger Cotes | Isaac Newton | Thomas Simpson

(dy)/(dx) = (dy)/(du)·(du)/(dx)

integral udv = uv - integral vdu | integral_a^budv = [uv]_a^b - integral_a^bvdu

lim(f(x)/(g(x))) = lim(f'(x)/(g'(x)))

d/(dx)(f(x) g(x)) = f(x) g'(x) + g(x) f'(x)

d/(dx)(f(x)/g(x)) = (g(x) f'(x)-f(x) g'(x))/([g(x)])^2

integral_(x_0)^(x_2)f(x)dx≈h(f_0 + 4f_1 + f_2)/3

integral_(x_0)^(x_1)f(x)dx≈h(f_0 + f_1)/2

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