## 2.1 Calculus

#### 2.1.1 The differentiation rules

Let f, g, h be differentiable and suppose

u = g(x) and v = h(x)

The constant rules

If y = cu = cg(x) where c Î R then
dy/dx = c du/dx = cg'(x)

The sum rule

If y = u + v = g(x) + h(x) then
dy/dx = du/dx + dv/dx = g'(x) + h'(x)

The product rule / Produktregelen

if y = u * v = g(x) h(x) then
dy/dx = du/dx * v + u * dv/dx = g'(x) h(x) + g(x) h'(x)

The quotient rule

If y = u/v = g(x) / h(x) where h(x) != 0 then
dy/dx = [ g'(x) h(x) - g(x) h'(x) ] / [h(x)]^2

The chain rule / Kjerneregelen

If y = f(u) = f(g(x)) and u = g(x) then
dy/dx = dy/du * du/dx = f''(g(x))*g'(x)

#### 2.1.1.1 Some special derivatives

(ln x)' = 1/x
(e^x)' = e^x
(cos x)' = -sin x
(sin x)' = cos x
(tan x) ' = sec^2 x = 1 / (cos^2 x)
(arctan x)' = 1 / (1 + x^2)
(arcsin)' = 1 / sqrt( 1 - x^2 )

#### 2.1.2 The laws of logarithms

x>0, y>0

log x + log y = log(xy)
log x - log y = log (x/y)
p log x = log (x^p)

#### 2.1.3 Newton's method

Let [a, b] be an interval containing just one solution of the equation f(x) = 0, where f ' is bounded, differentiable and positive on [a, b] and f '' is bounded and positive on [a, b]. The, for any x1Î [a, b], with f(x1) > 0, the sequence x1, g(x1), g(g(x1)),... converges to the solution of f(x) = 0 in [a, b], where

g(x) = x - [ f(x) / f '(x) ]

Here is a bit of code witch performs Newton's method:
(from Mørken)

```double x0, xp, x, e;
int n;
xp = x0; e = maxint; n = 1;
while (e>eps & n<=nmax) {
x = xp - f(xp)/Df(xp);
e = abs(x-xp);
n = n + 1;
}

```

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