 ### Calculus

1. Some useful derivatives:

1. d (c xn)/dx = c n xn - 1
2. d (sin x)/dx = cos x
d (cos x)/dx = - sin x
3. d (ef(x))/dx = ef(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:
ftotal = 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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