A.9  Calculus

f 1.

Derivative product rule: $(f(x)g(x))\prime =f(x)g\prime (x)+f\prime (x)g(x)$

f 2.

Derivative of a rational function

$$\left(\frac{f(x)}{g(x)}\right)\prime =\frac{f\prime (x)g(x)-f(x)g\prime (x)}{g^2(x)}$$

(A.22)

f 3.

Integration by parts:

$$\int \; <!--\; nolimits\; -->f(x)g\prime (x)\mathit{dx}=f(x)g(x)-\int \; <!--\; nolimits\; -->f\prime (x)g(x)\mathit{dx}$$

(A.23)

f 4.

Derivative of an exponential: $(e^{f(t)})\prime =e^{f(t)}f\prime (t)$

f 5.

Integral of an exponential: $\int (e^{f(t)})\mathit{dt}=\frac{e^{f(t)}}{f\prime (t)}$