Published on 11 Nov 2020
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Review of differentiation and integration rules from Calculus I and II
for Ordinary Differential Equations, 3301
General Notation:
a, b, m, n, C are non-specific constants, independent of variables
e, π are special constants e= 2.71828 ···,π= 3.14159 ···
f, g, u, v, F are functions
fn(x) usually means [f(x)]n, but f−1(x) usually means inverse function of f
a(x+y) means atimes x+y, but f(x+y) means fevaluated at x+y
fg means function ftimes function g, but f(g) means output of gis input of f
t, x, y are variables, typically tis used for time and xfor position, y is position or output
′,′′ are Newton notations for first and second derivatives.
Leibnitz notations for first and second derivatives are d
dx and d2
dx2or d
dt and d2
dt2
Differential of xis shown by dx or ∆xor h
f′(x) = lim
∆x→0
f(x+ ∆x)−f(x)
∆x, derivative of fshows the slope of the tangent line, rise
over run, for the function y=f(x) at x
General differentiation rules:
1a- Derivative of a variable with respect to itself is 1. dt
dt = 1 or dx
dx = 1.
1b- Derivative of a constant is zero.
2- Linearity rule (af +bg)′=af ′+bg′
3- Product rule (fg)′=f′g+fg′
4- Quotient rule µf
g¶′
=f′g−fg′
g2
5- Power rule (fn)′=nfn−1f′
6- Chain rule (f(g(u)))′=f′(g(u))g′(u)u′
7- Logarithmic rule (f)′= [eln f]′
8- PPQ rule (fngm)′=fn−1gm−1(nf′g+mfg′), combines power, product and quotient
9- PC rule (fn(g))′=nfn−1(g)f′(g)g′, combines power and chain rules
10- Golden rule: Last algebra action specifies the first differentiation rule to be used
Function-specific differentiation rules:
(un)′=nun−1u′(uv)′=uvv′ln u+vuv−1u′
(eu)′=euu′(au)′=auu′ln a
(ln(u))′=u′
u(loga(u))′=u′
ulog a
(sin(u))′= cos(u)u′(cos(u))′=−sin(u)u′
(tan(u))′= sec2(u)u′(cot(u))′=−csc2(u)u′
(sec(u))′= sec(u) tan(u)u′(csc(u))′=−csc(u) cot(u)u′