(d/dx)[C] = 0
(d/dx)[x] = 1
(d/dx)[Cu] = Cu’
(d/dx)[u ± v] = u’ ± v’
(d/dx)[u * v] = u’ * v’
(d/dx)[u / v] = (vu’ - u v’)/v^(2) .
(d/dx)[u^(n) ] = n*(u^(n-1)) * u’ (Chain Rule)
(d/dx)[Ln u] = u’ / u
(d/dx)[Log a
u] = u’/(Ln a) u
(d/dx)[e ^(u) ] = (e ^(n)) * u’
(d/dx)[a ^(u) ] = (Ln a)*(a ^(u)) * u’
/// Variable raised to a Variable ///
y = x ^(x)
Þ Ln y = Ln
x ^(n) Þ
Ln y = x Ln x Þ e ^(x Ln x) ^(u) Þ
y ’ = e ^(x Ln x) [x (1/x) + Ln x] Þ
y ’ = x ^(x) * (Ln x +
1)
/////= Integration =////////
ò e^(u) du = e^(u) + C
or
ò e^(u) u’ dx = e^(u) + C
other than e
5= e^(Ln 5) ,y = 5, Ln
y = Ln 5
Therefore y = e^(Ln 5)
/////= Integration =////////
ò ( e^(u)) du = ( e^(u)) + C
ò (a^(x) ) dx = (1/ (Ln a)) * (a^(x) ) + C
ò (a^(u)) du
= (1/ (Ln a)) * ( a^(u) ) + C
ò ( a^(u)) u ’dx
= (1/ (Ln a)) * ( a^(u) ) + C
ò ( u^(n)) du = (( u^(n)) / (n+1) ) + C, n ¹
-1
ò ( u -1) du or ò(1/u) du = Ln | u | + C
ò ( a^(x)) du = (1/ (Ln a)) * (a^(x)) + C
/////=
Population =////////
Rate
of change of Population is Proportional to Population.
(dP/ dt) = k * P
ò(1/P)dP = ò k dt
Ln P = k * t + C1
P = e^(kt + c1) = e^(kt
) e^( c1)
= Ce^(kt ) .
P = Ce^(kt )
.
P(0) =
Ce(0) is Population at zero
time
/////= Temperature =/////
Newton’s Cooling Law
(dT/ dt)= k(T – Ts)
k = Proportional, Ts= Surrounding Medium
Temperature.
(dT/ dt)= k(T – Ts) Þ
(1/(T
– Ts))dT = k dt Þ
Ln|(T – Ts)| = k*t + C1 Þ
|(T – Ts)|= e^(k*t +c1) =
e^(k*t) + e^(c1)
Þ C * e ^(k*t)
Final
formulas:
|(T – Ts)|= C * e ^(k*t)
T(t)
= Ts + C * e ^(k*t)
/////= e is inverse of Ln =/////
e= lim( xà∞ ) (1+(1/x))x ≈ 2.71828182846..
/////= Diff of Log =/////
(d/dx)[Log b x] = 1/(xLn
b)
where
y= b^(x),, b>0,b≠1 à
Lny=Lnb^(x)=xLnb
// === Diff of Number raised to x ===//
(d/dx) (b ^(x)) = (b ^(x))Ln(b)
//
=== If the Formulas are not above…. Go to this
link ===//
Exponential
Growth & Decay Introduction –Click this link
Derivative of
Trigonometric Functions and their Inverses
Verbal
Definition of Natural Log Function
.