(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) Ln y = Ln x ^(n)

Ln y = x Ln x e ^(x Ln x) ^(u)

y = e ^(x Ln x) [x (1/x) + Ln x]

y = x ^(x) * (Ln x + 1)

 

 

/////= Integration =////////

e^(u) du = e^(u) + C or

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 =////////

( e^(u)) du = ( e^(u)) + C

(a^(x) ) dx = (1/ (Ln a)) * (a^(x) ) + C

(a^(u)) du = (1/ (Ln a)) * ( a^(u) ) + C

( a^(u)) u dx = (1/ (Ln a)) * ( a^(u) ) + C

( u^(n)) du = (( u^(n)) / (n+1) ) + C, n -1

( u -1) du or (1/u) du = Ln | u | + C

( a^(x)) du = (1/ (Ln a)) * (a^(x)) + C

 

 

/////= Population =////////

Rate of change of Population is Proportional to Population.

(dP/ dt) = k * P

(1/P)dP = 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 =/////

Newtons Cooling Law

(dT/ dt)= k(T Ts)

k = Proportional, Ts= Surrounding Medium Temperature.

(dT/ dt)= k(T Ts)

(1/(T Ts))dT = k dt

Ln|(T Ts)| = k*t + C1

|(T Ts)|= e^(k*t +c1) = e^(k*t) + e^(c1)

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( x∞ ) (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 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 ===//

S.O.S Calculus index table

Exponential Growth & Decay Introduction Click this link

 

Derivative of Trigonometric Functions and their Inverses

 

Verbal Definition of Natural Log Function

 

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