Calculus Some useful derivatives: d (c xn)/dx = c n xn - 1 d (sin x)/dx = cos x d (cos x)/dx = - sin x d (ef(x))/dx = ef(x) df/dx d (ln x)/dx = 1/x d (f(x) + g(x))/dx = df/dx + dg/dx d (f(x) g(x))/dx = (df/dx) g(x) + f(x) (dg/dx) d f(g(x))/dx = df(g)/dg * dg(x)/dx d (∫ f(x) dx)/dx = f(x) ∫ df(x)/dx dx = f(x) Let f and g be functions related by an equation. Then the fractional change in g Δ g / g = (1 / g) Δ f / (df/dg). Any quantity f defined by integrating over contributions from infinitesimal pieces is additive: ftotal = f1 + f2. If u is a function only of v, (du/dv) * dv = du. Some useful integrals: ∫ √ (x2 + a) dx = (x √ (x2 + a) + a ln (x + √ (x2 + a))) / 2 ∫ dx / √ (x2 + a) = ln (x + √ (x2 + a)) ∫ dx / √ (x2 + a)3 = x / (a √ (x2 + a)) ∫ cos2 (a x) dx = ∫ sin2 (a x) dx = x / 2 + sin (2 a x) / (4 a) To integrate around a circle of radius r centered at the origin, use dx = -r sin θ dθ and dy = r cos θ dθ To solve the differential equation dx/dt = f(x) g(t), use separation of variables: dx / f(x) = g(t) dt and integrate the left side with respect to x and the right side with respect to t. Be sure to add a constant of integration. We will be interested in the following types of quantities: infinity finite numbers infinitesimal numbers zero
∫ df(x)/dx dx = f(x)
Δ g / g = (1 / g) Δ f / (df/dg).
ftotal = f1 + f2.
dx = -r sin θ dθ and dy = r cos θ dθ
dx / f(x) = g(t) dt
©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included. Please send comments or suggestions to the author.
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