EXPONENTIAL FUNCTIONS
197
Here, we designated 1 + C as the new arbitrary constant C x .
24.28
24.29
24.30
24.31
24.32
24.33
24.34
24.35
24.36
24.37
24.38
[Here, we have used the fact that D x (a") = (In a)a*.] Then
Then f x
3 e'*' dx = - H e" du = - \e" + C = - le~*' + C.
In Problems 24.30-24.39, find y'.
By implicit differentiation,
By implicit differentiation, sec
2 e"
x • e
y * -(y' -l) = 2x.
Thus, (l + x*)-e
y ~*-(y'-l) = 2x, y' = 1 4
Note that sec
2 e^* = 1 + tan
2 e^* = 1 + x
4
.
By implicit differentiation,
Use logarithmic differentiation. In y = sin x • In 3. [Here, we use the law ln(a
fr ) = bin a]. Hence,
So,
So, (!/>>)/= | (In 2)e*, y'=i(«n2)«*(V2)''.
J*
2 2*
3 Let « = 2
Jt3
, d« = ln2-2
J(3 -3x
2 dx.
Jjc
3 e^
4 ^.
Let M = -*
4
, dM = -4x
3 dx.
e
y = ^ + In x.
tan e
y ~* = x
2 .
e
lly + e
y = 2x.
X
2 + e*
y + y
2 = l.
2x + e*
y (xy'+y) + 2yy' = Q, y'(xe*
y + 2y) =-2x - ye*
y , y'=
sin x = e
y .
cosx = e
y y', y' =
y = y
ia
*.
(Hy)y' = (In 3)(cos x), y' = (In 3)(cos ^)(3
sin *).
y = (V2f.
ln> = e
j: -lnV2=|(ln2K.
y-*"'.
In y = In x • In x = (In x)
2
.
y = (ln*)"".
In >> = In x • In (In AC). So,
In
In
In
In
In