Analysis includes (according to the Oxford English Dictionary, OED) 'ancient analysis, the proving of a proposition by resolving it into simpler propositions already proved or admitted'; and 'modern analysis, the resolving of problems by reducing them to equations'. Analysis includes calculus, which is (according to OED) 'a system or method of calculation, "a certain way of performing mathematical investigations and resolutions" (Hutton); a branch of mathematics involving or leading to calculations, as the differential, integral calculus, etc.'. Isaac Newton & Gottfried Leibniz (the latter's system being more standard/advanced) started analysis, in which one studies thing such as series, sequences, summations, and more advanced operations that seem to be like algebra, except that they involve infinity and infinitesimality. Analysis, like geometry, depends on points, and higher analysis includes multivariable calculus, various differential equations subjects, numerical analysis, engineering analysis, etc., and abstract analysis including real & complex analysis. Analysis leads to understanding the secrets of the universe, and so is used by physicists to try to explain the universe, except that they only use the parts of analysis that they feel like using and try to ignore and get rid of the rest. Unfortunately, this is one of the reasons the way analysis has been taught has been changed with some vague and irrelevant ideas now being used. Only good mathematicians really know the secrets of the universe.
Mathematicians should know as much analysis as they can, not just to know the secrets of the universe, but because it is applied in almost all science, which is useful and helps advance civilization.
The basic form of differentiation and integration is below (but one should study the fundamental theorem of calculus before using these).
derivative (d/dx) of a variable, x, to a power, n: d/dx xn+c=nxn-1+0c
integral (∫) of a polynomial including a constant c: ∫ xn+c dx=(xn+1)/(n+1)+cx + c2
Most calculus instructors claim when you do 0*c in a derivative, that permanently gets rid of the constant. However, according to writers such as of The God Series (on philosophical mathematics) if every function has its derivative, there should also be integrals to get to every function, including ones with constants. It is not normally done, but just keep track of the constants if you ever need to (i.e. not in a standard math class, but there may be applications in new, idealist/mentalist/spiritualist/(Neo)Pythagorean-(Neo)Platonic-Leibnizian mathematics in the future).
Euler's constant (e) is an irrational number that when raised to the power of x, as ex, it has an unusual case.
A series of numbers that has exponents is a power series, and after trigonometry, one defines power series for sine & cosine waves. These series sum to Euler's Formula (EF,) The God Equation (GE.)
eix=cos x+i sin x
EF/GE defines all possible two-dimensional waves. There is a three-dimensional version, defining all possible waves in the material world (but from the view of either space or time) and the six-dimensional version is the Baker-Campbell-Hausdorff Formula (using three imaginary numbers that appear in temporal physics,) which is the secret of the universe, the only overall basis necessary to defining all of reality*.
*Hockney, Mike. The God Equation. Hyperreality Books, 2012 (also available on Lulu.com, Smashwords.com, and from commercial ebook stores.)