(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(TTs)

k = Proportional, Ts= Surrounding Medium Temperature.

(dT/ dt)= k(T – Ts) Þ

(1/(TTs))dT = k dt Þ

Ln|(TTs)| = k*t + C1     Þ

|(TTs)|= e^(k*t +c1) = e^(k*t) + e^(c1)

Þ   C * e ^(k*t)

Final formulas:

|(TTs)|= 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)

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