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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:
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
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
3> if Δ=b2-4ac < 0 then the solutions are complex and the general solution of the differential equation is
where r1,2=k±il=(-b/2a)±i[√b2-4ac /2a]
