Differentiate with respect to x 1. y H xL = I2x 3 + 5MI7x 4 + 3M Out[16]= Tick to show solution By using the formula for the derivative of a product we obtain uH xL = 2x 3 +5, u x = 6x 2 vH xL = 7x 4 +3, v x = 28 x 3 y x = 28 H2x 3 + 5L x 3 + 6 H7x 4 + 3L x 2 or y x = 2x 2 H49 x 4 + 70 x + 9L 2. yH xL = 2 ª x x 5 Out[48]= Tick to show solution By using the quotient rule we obtain uH xL = 2ª x , u x = 2 ª x vH xL = x 5 , v x = 1 5x 45 y x = 2 ª x 5x 45 + 2 ª x x 5 or y x = 2 ª x H5x+1L 5x 45 3. yH xL = lnH xLI12 x 3 - 4M
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Transcript
ü Differentiate with respect to x
1. y HxL = I2 x3
+ 5M I7 x4
+ 3M
Out[16]=
Tick to show solution
By using the formula for the derivative of a product we obtain
uHxL = 2x3+5,
„ u
„ x
= 6 x2
vHxL = 7x4+3,
„ v
„ x
= 28 x3
„ y
„ x
= 28 H2 x3 + 5L x
3 + 6 H7 x4 + 3L x
2
or
„ y
„ x
= 2 x2 H49 x
4 + 70 x + 9L
2. yHxL =2 ‰x
x5
Out[48]=
Tick to show solution
By using the quotient rule we obtain
uHxL = 2‰x,
„ u
„ x
= 2 ‰x
vHxL = x5
,
„ v
„ x
=1
5 x4ê5
„ y
„ x
=2 ‰x
5 x4ê5 + 2 ‰x
x5
or
„ y
„ x
=2 ‰x H5 x+1L
5 x4ê5
3. yHxL = lnHxL I12 x3 - 4M
Out[17]=
Tick to show solution
By using the product rule we obtain
uHxL = ln x,
„ u
„ x
=1
x
vHxL =12x3-4,
„ v
„ x
= 36 x2
„ y
„ x
=12 x
3-4
x+ 36 x
2logHxL
4. f HxL = Ix2+ 3M2
Out[33]=
Tick to show solution
By using the chain rule we get
uHxL = x2+ 3,
„ u
„ x
= 2 x
f HuL = u2,
„ f
„ u
= 2 u
„ f
„ x
= 4 x Hx2 + 3L
5. yHxL =6 x
2 - 4 x + 9
ln HxL
Out[13]=
Tick to show solution
By using the quotient rule we obtain
uHxL = 6x2-4x+9,
„ u
„ x
= 12 x - 4
vHxL = lnHxL,„ v
„ x
=1
x
„ y
„ x
=
1
HlnHxLL2H12 x - 4L ln HxL - I6 x
2- 4 x + 9M x
-1=
12 x-4
logHxL -6 x
2-4 x+9
x log2HxL
6. y HxL = H2 x + 4L4
2 | Numerical Integration
Out[32]=
Tick to show solution
By using the chain rule we get
uHxL = 2x + 4,
„ u
„ x
= 2
f HuL = u4,
„ f
„ u
= 4 u3
„ f
„ x
= 8 H2 x + 4L3
7. y HxL = H7 -6 xL2
Out[31]=
Tick to show solution
By using the chain rule we get
uHxL =7 - 6x,
„ u
„ x
= -6
f HuL = u2,
„ f
„ u
= 2 u
„ f
„ x
= -12 H7 - 6 xL
8. yHxL = 7 ‰xx
3
Out[19]=
Tick to show solution
By using the product rule we get
uHxL = 7‰x,
„ u
„ x
= 7 ‰x
vHxL = x3
,
„ v
„ x
=1
3 x2ê3
„ y
„ x
=7 ‰x
3 x2ê3 + 7 ‰x
x3
or
„ y
„ x
=7 ‰x H3 x+1L
3 x2ê3
9. y HxL =2 x
5
x + 1
Numerical Integration | 3
Out[8]=
Tick to show solution
By using the formula for the derivative of a quotient we obtain