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Example text

S" +1 sinwt n! +a)" + 1 w s2 +w2 coswt s s2 +w2 sin(wt +cp) ssincp+wcos(t)} =,n F(s) - sn - 1 F(O) """",n- 2 F'(O) L{F'(t)} - sF

COS -I :: -'- 1/(a 2 - x 2 )Y. tan -I af(x 2 + a 2 ) a :: sinhx coshx tanhx a coshx sinhx sech 2 x Product and quotient rules where du dv d -(uv)=u- +v- _! )= v ~- u d;; v2 v Leibnitz theorem Allowing u

and v

to denote the pth derivatives of u and v with respect to x, then n(n- 1) dn u (uv)=uv+nu+ - 2! dxn + n(n- 1Xn- 2) uv 3! Function of a [unction If y = f(u ), where u = cp(x ), then dy dy du -=-·dx du dx Maximum and minimum values off(x) A curve has a maximum value at a point if, at that point, d2 y dy dx =0 and dx 2 has a negative value.

J sin a = sinh ja sinja = j sinh a cos a = cosh ja cos ja =cosh a j tan a= tanhja tanja =j tanh a DIFFERENTIAL CALCULUS The differential coefficient (the derivative) of y with respect to x may be expressed in several ways, including DETERMINANTS dy d dx, dxy, Dy, Dxy,y , The solution of the simultaneous equations a 1 x +b 1y =A a2x + b2Y =B is given by 1 y -=-=Llx Ll y Ll X where I Llx = I~X:~ Lly = I=~ ~ I =a1B- a2A =Ab2 -Bb1 dy y dx axn aekx sin(ax +b) cos(ax +b) tan x cot x 1n x 1n(ax +b) sin -I The solution ofthe simultaneous equations a1x + b1y + c1z =A a2x + b 2y + c 2z =B a3x + b3y + c3z = C is given by :: a anxn-l akekx a cos(ax +b) -a sin(ax +b) sec 2 x - cosec 2 x 1/x a ax +b 1/(a 2 - x 2)Y.

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