CHAPTER 24
Exponential Functions
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
24.10
24.11
24.12
24.13
Evaluate e '"'.
In e "
: = — x by virtue of the identity In e" = a.
Find(e
2 )'
n *.
Evaluate (3e)
ln '.
Evaluate e
1
'"".
Findln(e*/:t).
In Problems 24.7-24.16, find the derivative of the given function.
195
for any real number r.]
[In like manner, D,(xr) =
By the quotient rule,
e" In x.
By the product rule, D x (e' In x) = e" • D,(ln x) + In jc • D x (e") = e* •
tan e".
By the chain rule, D,(tan e') = sec
2 e" • D x (e") = sec
2 e' • e' = e* sec
2 e".
By the chain rule, D x (e
mx ) = e
cos " • D x (cosx) = e
0 ** • (-sin x) = -e
cos * sin*.
By the chain rule, D,(«
l ") = «"'' °,d^) = «"' ' (~1^
2 ) = ~e"Vx
2 .
Evaluate In e *.
«-'»' = *'" <"-> = !/*.
S* \ln A- _ ^ ln3e\tn^M/ln3+l\lnAr_ /-'" Jr\ln3 + l __ In 3+1
e
1 -
tn ' = e
1 ela * = e/e
l '
l * = e/x.
In (e
A /x) = In e* — In x = x - In x. We have used the identities In (u/u) = In u - In f and \ne" = u.
e '.
By the chain rule, D f (e *) - e'* • D x (-x) = e *-(-\)=-e *. Here, we have used the fact that
D u (e") = e".
e
1 ".
e
cos *
e'/x.
+ In x • e' = e*
1
x
w .
D x (x") = D,(e" '"') = e"'"* • D t (ir In x) = e"
ln *
rx"
{
)'"- = ( ")2 = jc2. Here, we have used the laws 0")" = *"" and eln " = u.