1. Some useful derivatives:
   1. d (c xⁿ)/dx = c n x^{n - 1}
   2. d (sin x)/dx = cos x
      d (cos x)/dx = - sin x
   3. d (e^f(x))/dx = e^f(x) df/dx
      d (ln x)/dx = 1/x
   4. d (f(x) + g(x))/dx = df/dx + dg/dx
   5. d (f(x) g(x))/dx = (df/dx) g(x) + f(x) (dg/dx)
   6. d f(g(x))/dx = df(g)/dg * dg(x)/dx

2. d (∫ f(x) dx)/dx = f(x)

   ∫ df(x)/dx dx = f(x)

3. 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).

4. Any quantity f defined by integrating over contributions from infinitesimal pieces is additive:

   f_total = f₁ + f₂.

5. If u is a function only of v, (du/dv) * dv = du.
6. Some useful integrals:
   1. ∫ √ (x² + a) dx = (x √ (x² + a) + a ln (x + √ (x² + a))) / 2
   2. ∫ dx / √ (x² + a) = ln (x + √ (x² + a))
   3. ∫ dx / √ (x² + a)³ = x / (a √ (x² + a))
   4. ∫ cos² (a x) dx = ∫ sin² (a x) dx = x / 2 + sin (2 a x) / (4 a)

7. To integrate around a circle of radius r centered at the origin, use

   dx = -r sin θ dθ and
   dy = r cos θ dθ

8. 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.
9. We will be interested in the following types of quantities:
   1. infinity
   2. finite numbers
   3. infinitesimal numbers
   4. 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.