A: TABLE OF BASIC DERIVATIVES - University of Calgary in ...

1 A: TABLE OF BASIC DERIVATIVESLetu=u(x)be a differentiable function of the independent variablex, that isu (x)exists.(A) The Power Rule :Examples :ddx{un} =nun ddx{(x3+4x+1)3/4} =34(x3+4x+1) 1/4.(3x2+4)ddx{u} = ddx{2 4x2+7x5} =12 2 4x2+7x5( 8x+35x4)ddx{c} =0 ,cis a constantddx{ 6} =0 , since is a constant.(B) The Six Trigonometric Rules :Examples :ddx{sin(u)} =cos(u).u ddx{sin(x3)} =cos(x3). 3x2ddx{cos(u)} = sin(u).u ddx{cosx)} = sin(x).12xddx{tan(u)} =sec2(u).u ddx{tan{5x2)} =sec2(5x 2).( 10x 3)ddx{cot(u)} = csc2(u).u ddx[cot{sin(2x)}] = csc2{sin(2x)}. 2cos(2x).ddx{sec(u)} =sec(u)tan(u).u ddx{sec(4x)} =sec(4x)tan(4x).14x 3/4ddx{csc(u)} = csc(u)cot(u).u ddx{csc(8x 7)} = csc(8x 7)cot(8x 7). 8(C) The Six Hyperbolic Rules :Examples :ddx{sinh(u)} =cosh(u).u ddx{sinh(3x)} =cosh(3x).13x 2/3ddx{cosh(u)} =sinh(u).u ddx{cosh(sec(x)} =sinh{sec(x)}. sec(x)tan(x)ddx{tanh(u)} =sech2(u).}

2 U ddx[tanh{x3+sin(x2)}] =sech2{x3+sin(x2)}.(3x2+2xcos(x2))ddx{co th(u)} = csch2(u).u ddx{coth(1x+2x)} = csch2(1x+2x).( 1x2+2)ddx{sech(u)} = sech(u)tanh(u).u ddx{sech{9x)} = sech(9x)tanh(9x). 9ddx{csch(u)} = csch(u)coth(u).u ddx[csch{sinh(3x)}] = csch{sinh(3x)}coth{sinh(3x)}. 3cosh(3x)(D) The Exponential&Logarithmic Rule :Examples :ddx{eu} = ddx{e x3} =e x3.( 3x2)ddx{ln|u|} =u uddx{ln|x3+5x+6|} =3x2+5x3+5x+6ddx{au} = (a).u ,a R,a>0 ,a 1ddx{2sec(x)} =2sec(x).ln(2). sec(x)tan(x)ddx{loga|u|} =1ln(a)u u,a R,a>0 ,a 1ddx{log4|tan(x) |} =1ln(4)sec2(x)tan(x)(E) The Six Inverse Trigonometric Functions :Examples :ddx{sin 1(u)} =u 1 u2ddx{sin 1(4x2)} =8x1 16x4ddx{cos 1(u)} = u 1 u2ddx{cos 1(3x)} = 31 9x2ddx{tan 1(u)} =u 1+u2ddx{tan 1(x)} =12x1+x=12x(1+x)ddx{cot 1(u)} = u 1+u2ddx{cot 1(ex)} = ex1+e2xddx{sec 1(u)} =u |u|u2 1ddx[sec 1(x4)} =4x3|x4|x8 1=4x3x4x8 1ddx{csc 1(u)} = u |u|u2 1ddx{csc 1(2x)} = 2|2x|4x2 1= 1|x|4x2 1(F) The Inverse Hyperbolic Functions :Examples :ddx{sinh 1(u)} =u 1+u2ddx{sinh 1(ln(x)} =1/x1+ln2(x)ddx{cosh 1(u)} =u u2 1ddx{cosh 1(5x)} =525x2 1ddx{tanh 1(u)} =u 1 u2ddx{tanh 1(2x)} = 2x21 4x2= 2x2 4(G) The Product and Quotient Rules :Examples :ddx{uv} =u v+uv ddx{x3ln(5x+1)} =3x2ln(5x+1) +x355x+1ddx{ku} =ku ,kis a constantddx{x34} =14ddx{x3} =14.]

3 3x2=3x24ddx{uv} =u v uv v2ddx{tan(2x)x3} =2sec2(2x).x3 tan(2x).3x2x6In TABLE above it is assumed thatu=u(x)andv=v(x)are differentiable functionsB: TABLE OF BASIC INTEGRALSLetr,a,b, and R,r 1 ,a 0 , and >0.(A) The Power Rule :Examples : (ax+b)rdx=(ax+b)r+1a(r+1)+C x 5dx= 14x 4+C, (3x 1) 2dx=(3x 1) 1 3+C dx= 1dx=x+C 7dx=7 dx=7x+C 1ax+bdx=2aax+b+C 1x+4dx=2x+4+C.(B) The Six Trigonometric Rules :Examples : sin(ax+b))dx= 1acos(ax+b) +C sin(9x 2)dx= 19cos(9x 2) +C cos(ax+b)dx=1asin(ax+b) +C cos(3x)dx=13sin(3x) +C tan(ax+b)dx=1aln|sec(ax+b)|+C tan(5w 1)dw=15ln|sec(5w 1)|+C cot(ax+b)dx=1aln|sin(ax+b)|+C cot(1 7u)du= 17ln|sin(1 7u)|+C sec(ax+b)dx=1aln|sec(ax+b) +tan(ax+b)|+C sec(3x)dx=13ln|sec(3x) +tan(3x)|+C csc(ax+b)dx=1aln|csc(ax+b) cot(ax+b)|+C csc(2t)dt=12ln|csc(2t) cot(2t)|+C(C) Additional Trigonometric Rules :Examples sec2(ax+b)dx=1atan(ax+b) +C sec2(2u/3)du=32tan(2u/3) +C csc2(ax+b)dx= 1acot(ax+b) +C csc2(w2)dw= 11/2cot(w2) +C= 2cot(w2) +C sec(ax+b)tan(ax+b)dx=1asec(ax+b) +C sec(3u)tan(3u)du=13sec(3u) +C csc(ax+b)cot(ax+b)dx= 1acsc(ax+b) +C csc(5x)cot(5x)dx= 15csc(5x) +C(D) The Six Hyperbolic Rules.

4 Examples sinh(ax+b)dx=1acosh(ax+b) +C sinh(2x 7)dx=12cosh(2x 7) +C cosh(ax+b)dx=1asinh(ax+b) +C cosh(2x5)dx=52sinh(2x5) +C tanh(ax+b)dx=1aln[cosh(ax+b)] +C tanh(2u)du=12ln[cosh(2u)] +C coth(ax+b)dx=1aln|sinh(ax+b)|+C coth(x+3)dx=ln|sinh(x+3)|+C sech(ax+b)dx=2atan 1(eax+b) +C sech(3x 6)dx=23tan 1(e3x 6) ++C csch(ax+b)dx=1aln|tanh(ax+b)/2|+C csch(10t)dt=110ln|tanh(5t)|+C(E) Additional Hyperbolic Rules :Examples sech2(ax+b)dx=1atanh(ax+b) +C sech2(4w)dw=14tanh(4w) +C csch2(ax+b)dx= 1acoth(ax+b) +C csch2(2u)du= 12coth(2u) +C sech(ax+b)tanh(ax+b)dx= 1asech(ax+b) +C sech(3x)tanh(3x)dx= sech(3x)3+C csch(ax+b)coth(ax+b)dx= 1acsch(ax+b) +C csch(x3)coth(x3)dx= 3csch(x/3) +C(F) Exponential/Logarithmic Rules :Examples : eax+bdx=1aeax+b+C e7xdx=17e7x+C k x+ dx=1aln(k).kax+b+C, 0<k R,k 1. 210x 17dx=110ln2210x 17+C 1ax+bdx=1aln|ax+b|+C 12x 3dx=12ln|2x 3|+C.(G) The Three Inverse Trigonometric Functions :Examples : 1 2 x2dx=sin 1(x ) +C 116 x2dx=sin 1(x/4) +C 1 2+x2dx=1 tan 1(x ) +C 13+x2dx=13tan 1(x3) +C 1x x2 2dx=1 sec 1(x ) +C,x> 1x x2 4dx=12sec 1(x2) +C,x>2.

5 (H) The Three Inverse Hyperbolic Functions :Examples : 1 2+x2dx=sinh 1(x ) +C 11+x2dx=sinh 1(x) +C 1x2 2dx=cosh 1(x ) +C 1x2 5dx=cosh 1(x/5) +C 1 2 x2dx=1 tanh 1(x ) +C,|x|< 136 x2dx=16tanh 1(x6) +C,|x|<6(I) The Fundamental TheoremsExamples : abf(x)dx=g(x)|x=ax=b=g(b) g(a) ee31xdx=ln|x| |x=ex=e3=ln(e3) ln(e) =3 1=2ddx{ u(x)v(x)F(t)dt=F(v(x)).v (x) F(u(x)).u (x)ddx{ xx2cos(t2)dt=cos(x4).2x cos(x2).1In TABLE above it is assumed that :(1)The functionf(x)is continuous on[a,b]and f(x)dx=g(x) +C.(2)The functionsu(x)andv(x)are differentiable and u(x)v(x)F(t) : BASIC TRIGONOMETRIC IDENTITIESGROUP(A):(i)tan( ) =sin( )cos( )(ii)cot( ) =cos( )sin( )(iii)sec( ) =1cos( )(iv)csc( ) =1sin( )GROUP(B):(i)cos2( ) +sin2( ) =1(ii)1+tan2( ) =sec2( )(iii)cot2( ) +1=csc2( )GROUP(C):(i)sin(2 ) =2 sin( )cos( )(ii)cos(2 ) =2cos2( ) 1(iii)cos(2 ) =1 2sin2( )(iv)cos2( ) =12[1+cos(2 )](v)sin2( ) =12[1 cos(2 )]GROUP(D)(i)sin( ) = sin( )(ii)cos( ) =cos( )(iii)tan( ) = tan( ).}}

6 GROUP(E)(i)cos( ) =cos( )cos( ) sin( )sin( )(ii)sin( ) =sin( )cos( ) cos( )sin( )GROUP(F)(i)cos( )cos( ) =12[cos( ) +cos( + )](ii)sin( )sin( ) =12[cos( ) cos( + )](iii)sin( )cos( ) =12[sin( ) +sin( + )]D:SPECIAL TRIGONOMETRIC EQUATIONS(i)sin(x) =0 x=n (ii)cos(x) =0 x=(2n 1)2 wherenis an integer :n=0, 1, 2, 3,..E:HYPERBOLIC FUNCTIONS(i)sinh(x) =12[ex e x](ii)cosh(x) =12[ex+e x](iii)tanh(x) =sinh(x)cosh(x)(iv)coth(x) =cosh(x)sinh(x)(v)sech(x) =1cosh(x)(vi)csch(x) =1sinh(x)(vii)cosh2(x) sinh2(x) =1(viii)1 tanh2(x) =sech2(x)(ix)coth2(x) 1=csch2(x)F:PROPERTIES OF LOGARITHMSL etxandybe positive real numbers.(i)ln(x) +ln(y) =ln(xy)(ii)ln(x) ln(y) =ln(xy)(iii)ln(xm) =mln(x).(iv)ln(ek) =k(v)eln(x)=x(vi)ln(1) =0 , ln(e) = :SPECIAL VALUES(i)sin(0) =0(ii)cos(0) =1(iii)tan(0) =0(iv)sinh(0) =0(v)cosh(0) =1(vi)tanh(0) =0(vii)sin(n ) =0 and cos(n ) = ( 1)n, provided that "n" is an