627 05' ENGINEERING FORMULÆ.
DIFFERENTIAL AND INTEGRAL CALCULUS—con tinned.
Differentiation of various Expressions of v, y, w, &g., functions of
_______________________the variable x.
If = d u then — = dx
Sum of several) functions .. j Product of two) functions .. f Product of more ) than two fnnc- > tions .. .. J Fraction ,. .. Power .. .. Fractional power Negative power Function of a) function .. j v + y - w &c. v.y v.y.w V yn n y™ y~n d v dy dw, T- + -T T~ &c- dx dx dx dy d v v r~ + y~r dx dx dy dv dw v.w. — + w.y — + v.y dx dx dx dy n.y«-1 — d x n !!__x dy m dx dy - nyn-i — dx du dy dy ' dx
Successive Differ kntiation. Successive differentiation is the process of differentiating successive differential coefficients of an original function.
Thus if u - a xs d u 1st coefficient zz — — 5 a d X Ä d2 u 2nd „ = —20 a d x2 d3 u 3rd coefficient = -y-^- — 60 ax2 d^ u 4th ,, = z=120ax dx* d3 u 5ih „ = = 120 a a*s
Th the numerators given in the example above d2 u, d3 u, $c., the indices are simply
the symbols of successive differentiation.
_ , du , Ja u ,
If u = a*; then —— = log.F a ,ax; -— = (log. a) 2 ax ;
d x c a x% c
d3 u , d* u , ’ ' ' [ ’ _
— = (log. a) 3 ax; -— = (log. a)»«1; &e.
dx3 e dx* e
where € = number whose hyp. log. is 1 =2*71828
„ du 1^2u 1 d3 u .1.2
If u — log. x; then — — 4—; ----—--------; r— — H------;
e dx X dx2 dx3 x3
d*u 1.2.3 d*u 12.3,4 „
--- -----------; -- = ----------; &C,
d x* x* dxS x&
^U . U •
J f ?( ■= sm. x; then •— 4- cos. x • - = — sin. x ;
dx dx2
dsu d*u , ds u . . .
--= — C08. 4-; ---- = 4- bin. x; -= -f- COSin. x ; &c.
dxs dx* dxs
2 s 2