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- Some useful derivatives:
- d (c x
^{n})/dx = c n x^{n - 1} - d (sin x)/dx = cos x

d (cos x)/dx = - sin x - d (e
^{f(x)})/dx = e^{f(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 (c x
- 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:
f

_{total}= f_{1}+ f_{2}. - If u is a function
*only*of v, (du/dv) * dv = du. - Some useful integrals:
- ∫ √ (x
^{2}+ a) dx = (x √ (x^{2}+ a) + a ln (x + √ (x^{2}+ a))) / 2 - ∫ dx / √ (x
^{2}+ a) = ln (x + √ (x^{2}+ a)) - ∫ dx / √ (x
^{2}+ a)^{3}= x / (a √ (x^{2}+ a)) - ∫ cos
^{2}(a x) dx = ∫ sin^{2}(a x) dx = x / 2 + sin (2 a x) / (4 a)

- ∫ √ (x
- 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

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©2010, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.