1) In Problems 1 through 4, given f(x) = 3x , g(x) = eX, h(x) = In x .Use Dr. Dai's Differential and Integration Table to find[f(x) – ln3 g(x) ] ' =A) x3X-1 -In3 (xeX-1)B) In3 (3x -ex)C) (3×ln3 -eX)D) (ex -In 3)2) Find (g(x) h(x))' = ,[ By Product Rule]C) ex(lnx+ eX)A) (ex Inx/ x)B) eX(1+x)D) eX(In x +1/x)3) Using Quotient Rule (-)'=C g-fg), find ()'= (3×/eX)' :-A) (3/e)X[In3-1]B) (3/e)x-1C) 3Xln3/exD) 3x-1x /ex4) Find [(h(x) )3 ]'=[(ln x) 3 ] ' = ,A) 3 In x2(In x)By Power RuleB) (In x) 2/xC) 3(In x) 2/xD) 3 x2/In x a +1Dr. Hong Dai's Formulae for Calculus or Higher Courses (MAT201-MAT410)*Chain Rule: F" (g(x)) =» f(g(x))g',Basic Rule: F"(x) = f(x).Poly.1F1:x(a+1) =x° (if a ±-1).1g(x)(a+) = g(x)ʻg', (if a # -1)Func.F2:(and F15) (In x) :== (if a = -1)(and F16) (In g(x)) = g(x)"g', =8', (if a = -1)g(x)OR (* ) = cOR (g(2)*) = ag (x)* g'.F4: (sin g(x)) = cos g(x)g',F6: (cos g(x)) =-sin g(x)g';F8: (tan g(x)) = sec² g(x)g',F10: (cot g(x)'F12: (sec g(x)) = sec g(x) tan g(x)g'.F14: (csc g(x)) =– csc g(x)cot g(x)g'.Trig.F3: (sinx)= cos xFunc.F5: (cosx)F7: (tanx) = sec2x=- sin xF9: (cotx) :=- csc? x=-csc? g(x)g',F11: (sec x) = sec x tan xF13: (csex) =%3D-csc x cot xLog orExp.Func.F15: (Inx)1=x =.F16: (In g(x)) = g(x)*g' =8(x)'%3D11F17: (log, x) :x InbF18: (log, g(x)) :g(x)(Inb)F19: (e*) = e-F21: (b) = b* InbF20: (es) = eg':F22: (br)) = (5ec2) Inb)g',F23: (sin x) =1//i-x?F25: (cos" 2) = -1//1–x²(F27: (tan-x) =1/(1+x*)F29: (cot" x) = -1/(1+x²)F31: (sec- x) =1/] x | Vz² -1]F33: (csex) = -1/[] × |Vx7 -1]F24: (sin" g(x) =[1//I-8(x)* lg':F26: (cos g(x) = -[1/1-g(x)* lg',F28: (tan- g(x)) = ([1/[1 + g(x)*])g'.F30: (cot" g(x) =-([1/[1 + g(x)*])g';F32: (sec" g(x) = {1/[I g(x)| /g(*)* –1]}g'.F34: (csc g(x)) = -{1/I g(x)|/g(x)* –1]}g'.F36: (sinh g(x)) = cosh g(x)g',F38: (cosh g(x)) = sinh g(x)g'.F40: (tanh g(x)) = sec h? g(x)g',Iny.Trig.Func.Hyper. F35: (sinh x) = cosh xF37: (cosh x) = sinh xF39: (tanh x) = sech²xFunc.*NotesF41:[x] = 1; F42: [C] = 0; F43:[x*]' = 2x; F44: [÷ sin(kx)] = cos(kx); F45: [cos(kx)] =-sin(kx);7: (x-1)e] =xe; F48:-k² +b²1-e [k sin(bx)– b cos(kx)]}= e. F47: [= e sin(bx);Product Rule: (fg) = f'g+ fg';Quotient Rule: (f/g) = (f'g- fg')/g²; Power Rule: (f) = of fFTC: f(2)dx = [ f(x)ix - F(*) = F(b) – F(a);Basic Rule: F(x)+C = [f(x)dIntegration by Parts: fdg = fg- gdfChain Rule: F(g(x))+C =* DERIVATIVES? Read from LEFT to RIGHT./ INTEGRALS? Read from RIGHT to LEFT (without prime + C)

Question

1) In Problems 1 through 4, given f(x) = 3x , g(x) = eX, h(x) = In x .Use Dr. Dai's Differential and Integration Table to find[f(x) – ln3 g(x) ] ' =A) x3X-1 -In3 (xeX-1)B) In3 (3x -ex)C) (3×ln3 -eX)D) (ex -In 3)2) Find (g(x) h(x))' = ,[ By Product Rule]C) ex(lnx+ eX)A) (ex Inx/ x)B) eX(1+x)D) eX(In x +1/x)3) Using Quotient Rule (-)'=C g-fg), find ()'= (3×/eX)' :-A) (3/e)X[In3-1]B) (3/e)x-1C) 3Xln3/exD) 3x-1x /ex4) Find [(h(x) )3 ]'=[(ln x) 3 ] ' = ,A) 3 In x2(In x)By Power RuleB) (In x) 2/xC) 3(In x) 2/xD) 3 x2/In x a +1Dr. Hong Dai's Formulae for Calculus or Higher Courses (MAT201-MAT410)*Chain Rule: F&#34; (g(x)) =» f(g(x))g',Basic Rule: F&#34;(x) => f(x).Poly.1F1:x(a+1) =x° (if a ±-1).1g(x)(a+) = g(x)ʻg', (if a # -1)Func.F2:(and F15) (In x) :== (if a = -1)(and F16) (In g(x)) = g(x)&#34;g', =8', (if a = -1)g(x)OR (* ) = cOR (g(2)*) = ag (x)* g'.F4: (sin g(x)) = cos g(x)g',F6: (cos g(x)) =-sin g(x)g';F8: (tan g(x)) = sec² g(x)g',F10: (cot g(x)'F12: (sec g(x)) = sec g(x) tan g(x)g'.F14: (csc g(x)) =– csc g(x)cot g(x)g'.Trig.F3: (sinx)= cos xFunc.F5: (cosx)F7: (tanx) = sec2x=- sin xF9: (cotx) :=- csc? x=-csc? g(x)g',F11: (sec x) = sec x tan xF13: (csex) =%3D-csc x cot xLog orExp.Func.F15: (Inx)1=x =.F16: (In g(x)) = g(x)*g' =8(x)'%3D11F17: (log, x) :x InbF18: (log, g(x)) :g(x)(Inb)F19: (e*) = e-F21: (b) = b* InbF20: (es) = eg':F22: (br)) = (5ec2) Inb)g',F23: (sin x) =1//i-x?F25: (cos&#34; 2) = -1//1–x²(F27: (tan-x) =1/(1+x*)F29: (cot&#34; x) = -1/(1+x²)F31: (sec- x) =1/] x | Vz² -1]F33: (csex) = -1/[] × |Vx7 -1]F24: (sin&#34; g(x) =[1//I-8(x)* lg':F26: (cos g(x) = -[1/1-g(x)* lg',F28: (tan- g(x)) = ([1/[1 + g(x)*])g'.F30: (cot&#34; g(x) =-([1/[1 + g(x)*])g';F32: (sec&#34; g(x) = {1/[I g(x)| /g(*)* –1]}g'.F34: (csc g(x)) = -{1/I g(x)|/g(x)* –1]}g'.F36: (sinh g(x)) = cosh g(x)g',F38: (cosh g(x)) = sinh g(x)g'.F40: (tanh g(x)) = sec h? g(x)g',Iny.Trig.Func.Hyper. F35: (sinh x) = cosh xF37: (cosh x) = sinh xF39: (tanh x) = sech²xFunc.*NotesF41:[x] = 1; F42: [C] = 0; F43:[x*]' = 2x; F44: [÷ sin(kx)] = cos(kx); F45: [cos(kx)] =-sin(kx);7: (x-1)e] =xe; F48:-k² +b²1-e [k sin(bx)– b cos(kx)]}= e. F47: [= e sin(bx);Product Rule: (fg) = f'g+ fg';Quotient Rule: (f/g) = (f'g- fg')/g²; Power Rule: (f) = of fFTC: f(2)dx = [ f(x)ix - F(*) = F(b) – F(a);Basic Rule: F(x)+C = [f(x)dIntegration by Parts: fdg = fg- gdfChain Rule: F(g(x))+C =* DERIVATIVES? Read from LEFT to RIGHT./ INTEGRALS? Read from RIGHT to LEFT (without prime + C)

Accepted Answer

1.) f(x)=3x, gx=ex, hx=lnx
fx-ln3gx'=f'x-ln3g'x                              =3x ln3-ln3 ex…