MATH1101 Lecture 8: reviewdifint

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Published on 11 Nov 2020
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 f1(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
Differential of xis shown by dx or ∆xor h
f(x) = lim
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)=fg+fg
4- Quotient rule µf
5- Power rule (fn)=nfn1f
6- Chain rule (f(g(u)))=f(g(u))g(u)u
7- Logarithmic rule (f)= [eln f]
8- PPQ rule (fngm)=fn1gm1(nfg+mfg), combines power, product and quotient
9- PC rule (fn(g))=nfn1(g)f(g)g, combines power and chain rules
10- Golden rule: Last algebra action specifies the rst differentiation rule to be used
Function-specific differentiation rules:
(un)=nun1u(uv)=uvvln u+vuv1u
(eu)=euu(au)=auuln a
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
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