USEFUL FORMULAS. ^^^ 



Gauss's formulas. 



cos i (a + /3) cos \ c = cos \ {a + b) sin I y, 

 sin i (a + /3) cos i ^ = cos \ {a - h) cos i y, 



cos \ia- ^)s\n\c — sin i (a + /^ sin \ y, 

 sin i (a — /3) sin i r = sin ^ {a - b) cos ^ y. 



5. Elementary Differential Formulas. 

 a. Algebraic. 



«, V, 7CI, . . . = variables subject to differentiation, 

 a, b, c, . . .■=■ constants. 



dia + u) = du, d{a u) = a du, 

 di^u -f r + a/ + ...) = ^/ « + ^^ 7v + ^/ w + ... , 



d{ll V) = U dv -\- V du, 



(du , dv . dw x_ \ 

 ^ + V + "^ + -"j'''^'^-"' 



'0)= 



d\_\— ^;2 — ^ j;2 



V du — u d%) du u dv 



(a -\r b u\ ^ .. .. a 



(a -\-b_u\ b h — a g 



^z/"= nv"-'^ dv, d\Jv = ^— r' 



^/^'' = «'• log a dv, de"-' = e'' dv 



(e = base of natural logarithms), 



d log v = dvfv. 



b. Trigonometric and inverse trigonometric. 



fl'sin X = cos X dx, dcos x = — sin .v dx, 



dtSLn X = sec'" .v dx, dcot x =. — cosec^ .r dx, 



dsec X = sec- x sin x dx, ^/cosec .v = — cosec^ .v cos .r dx. 



dlog sin :v =: cot x dx, dlog cos x = — tan a: ^.v. 



dx , dx 



dare sin x = ± , „' «arc cos .v = T / 



Vi — x^ VI — •* 



dx dx 



^arc tan .r = ^ . ^o, ^arc cot .r = — 711^.2* 



2 



