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′