1) In Problems 1 through 4, given f(x) = 3× , 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) C) (3×ln3 -eX) B) In3 (3x -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(ln x +1/x) 3) Using Quotient Rule (-)'=€'g-tg), find (-)' = (3×/eX)' = (3X/eX)' = _ %3D A) (3/e)X[In3-1] B) (3/e)X-1 C) 3×ln3/eX D) 3x-1x /ex E) Find [(h(x) )3]'=[(ln x) 3 ] ' =. A) 3 In x2(ln x) _By Power Rule C) 3(ln x) 2/x B) (In x) 2/x D) 3 x2/In x
1) In Problems 1 through 4, given f(x) = 3× , 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) C) (3×ln3 -eX) B) In3 (3x -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(ln x +1/x) 3) Using Quotient Rule (-)'=€'g-tg), find (-)' = (3×/eX)' = (3X/eX)' = _ %3D A) (3/e)X[In3-1] B) (3/e)X-1 C) 3×ln3/eX D) 3x-1x /ex E) Find [(h(x) )3]'=[(ln x) 3 ] ' =. A) 3 In x2(ln x) _By Power Rule C) 3(ln x) 2/x B) (In x) 2/x D) 3 x2/In x
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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-1
C) 3Xln3/ex
D) 3x-1x /ex
4) Find [(h(x) )3 ]'=[(ln x) 3 ] ' = ,
A) 3 In x2(In x)
By Power Rule
B) (In x) 2/x
C) 3(In x) 2/x
D) 3 x2/In x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72a720b9-38b2-4beb-ae1b-56a99a6fccf3%2Fef25834f-b580-4b73-8b13-61bfb127984d%2Fqor9xh7_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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-1
C) 3Xln3/ex
D) 3x-1x /ex
4) Find [(h(x) )3 ]'=[(ln x) 3 ] ' = ,
A) 3 In x2(In x)
By Power Rule
B) (In x) 2/x
C) 3(In x) 2/x
D) 3 x2/In x
![a +1
Dr. 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.
1
F1:
x(a+1) =x° (if a ±-1).
1
g(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 (* ) = c
OR (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 x
Func.
F5: (cosx)
F7: (tanx) = sec2x
=- sin x
F9: (cotx) :
=- csc? x
=-csc? g(x)g',
F11: (sec x) = sec x tan x
F13: (csex) =
%3D
-csc x cot x
Log or
Exp.
Func.
F15: (Inx)
1
=x =.
F16: (In g(x)) = g(x)*g' =
8(x)'
%3D
1
1
F17: (log, x) :
x Inb
F18: (log, g(x)) :
g(x)(Inb)
F19: (e*) = e-
F21: (b) = b* Inb
F20: (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 x
F37: (cosh x) = sinh x
F39: (tanh x) = sech²x
Func.
*Notes
F41:[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 +b²
1
-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 f
FTC: f(2)dx = [ f(x)ix - F(*) = F(b) – F(a);
Basic Rule: F(x)+C = [f(x)d
Integration by Parts: fdg = fg- gdf
Chain Rule: F(g(x))+C =
* DERIVATIVES? Read from LEFT to RIGHT./ INTEGRALS? Read from RIGHT to LEFT (without prime + C)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72a720b9-38b2-4beb-ae1b-56a99a6fccf3%2Fef25834f-b580-4b73-8b13-61bfb127984d%2Fwrb0d8f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:a +1
Dr. 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.
1
F1:
x(a+1) =x° (if a ±-1).
1
g(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 (* ) = c
OR (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 x
Func.
F5: (cosx)
F7: (tanx) = sec2x
=- sin x
F9: (cotx) :
=- csc? x
=-csc? g(x)g',
F11: (sec x) = sec x tan x
F13: (csex) =
%3D
-csc x cot x
Log or
Exp.
Func.
F15: (Inx)
1
=x =.
F16: (In g(x)) = g(x)*g' =
8(x)'
%3D
1
1
F17: (log, x) :
x Inb
F18: (log, g(x)) :
g(x)(Inb)
F19: (e*) = e-
F21: (b) = b* Inb
F20: (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 x
F37: (cosh x) = sinh x
F39: (tanh x) = sech²x
Func.
*Notes
F41:[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 +b²
1
-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 f
FTC: f(2)dx = [ f(x)ix - F(*) = F(b) – F(a);
Basic Rule: F(x)+C = [f(x)d
Integration by Parts: fdg = fg- gdf
Chain Rule: F(g(x))+C =
* DERIVATIVES? Read from LEFT to RIGHT./ INTEGRALS? Read from RIGHT to LEFT (without prime + C)
Expert Solution
Step 1
1.)
Hence, option (B) is the correct option.
Step by step
Solved in 4 steps
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