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## Derivatives of non natural logarithms and exponentials

Some useful calculations to keep in mind !!

> changing bases in logarithms: loga x = logb x / logb a
for base 10 and e: log x = ln x / ln 10
(log e = 1 / ln 10)

> derivative of exponential with other base:

d (ax) / dx = ax ln a

in the case of base 10: d (10x) / dx = 10x ln 10

> derivative of logarithm with other base:

d (loga x) / dx = d (ln x / ln a) / dx = 1 / ( x ln a)

so in the case of base 10: d (log x) / dx = 1 / ( x ln 10)

A useful online resource for calculations is Wolfram Alpha ‘s engine.

## Differential Equations -A

Solutions of a second order differential linear and homogeneous equation with constant coefficients:

a y” + b y’ + c y = 0 (a ≠ 0)

Solve the characteristic equation: a r2 + b r + c = 0
with general solutions r1,2=[(-b±√b2-4ac )/2a].

1> if Δ=b2-4ac > 0 then the solutions r1=r1 are real and
the general solution of the differential equation is

y = c1 er1x + c2 er2x

2> if Δ=b2-4ac = 0 then the solutions are real and equal (r1=r>2) and the general solution of the differential equation is

y = c1 er1x + c2 x er1x

3> if Δ=b2-4ac < 0 then the solutions are complex and the general solution of the differential equation is

y = c1 ekxcos(lx) + c2 ekxsin(lx)

where r1,2=k±il=(-b/2a)±i[√b2-4ac /2a]