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Mathematical Exercise on Natural Logarithms, Exercises of Mathematical logic

China Medical CollegeMathematical logic

The solution to mathematical exercise problem number 3, which involves finding the value of y1, y2, and y3 given certain equations that include natural logarithms.

Typology: Exercises

2019/2020

Uploaded on 02/12/2020

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EX 3. 試求 d
dx x
x
x
,d
dx
(
x
x
)
x
¿i>設y
1
=x
x
⇒ln y
1
=xln x⇒y
1
'
y
1
=ln x+1⇒y
1
'
=x
x
(
ln x+1
)
¿ii>設y
2
=x
x
x
⇒ln y
2
=x
x
ln x⇒y
2
'
y
2
=x
x
(
ln x+1
)
ln x+x
x
x
⇒y
2
'
=x
x
x
x
x
(
(
ln x
)
2
+ln x+1
x
)
¿iii>設y
3
=
(
x
x
)
x
⇒ln y
3
=x
2
ln x⇒y
3
'
y
3
=2xln x+x
⇒y
3
'
=
(
x
x
)
x
(
2xln x+x
)

Partial preview of the text

Download Mathematical Exercise on Natural Logarithms and more Exercises Mathematical logic in PDF only on Docsity!

EX 3. 試求

d dx x xx , d dx

( x

x

x ¿ i>設 y 1 =x x ⇒ ln y 1 =x ln x ⇒ y 1 ' y 1 =ln x + 1 ⇒ y 1 ' =x x ( ln x + 1 ) ¿ ii>設 y 2 =x xx ⇒ ln y 2 =x x ln x ⇒ y 2 ' y 2 =x x ( ln x + 1 ) ln x+ x x x ⇒ y 2 ' =x x x x x

( ln x ) 2 +ln x+

x )

¿ iii> 設 y 3 =( x

x

x ⇒ ln y 3 =x 2 ln x ⇒ y 3 ' y 3 = 2 x ln x+ x ⇒ y 3 '

=( x

x

x ( 2 x ln x+ x )