Chapter 9 Exponential and Logarithmic Functions指数函数和对数函数的导数
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Rule No.1: If
y ln x
, then
dy 1 dx x
This rule has a corollary that incorporates the Chain Rule and is actually a more useful rule to memorize: • Rule No.2: If
ln x3 3 ln x
1 3 f x 3 x x
Therefore,
Example 2: Find the derivative of f x ln 5x 3x 6 .
5 18 x f x 5x 3x
ln x log a x ln a
Review the unit on Prerequisite Mathemactics if this leaves you scratching your head. Anyway, because ln a is a constant, we can take the derivative and we get:
Chapter 9 Exponential and Logarithmic Functions, Part One
AP Math Bella
As with trigonometric functions, you'll be expected to remember all of the logarithmic and exponential functions you've studied in the past. If you're not sure about any of this stuff. review the unit on Prerequisite Mathematics. Also, this is only part one of our treatment of exponents and logs. Much of what you need to know about these functions requires knowledge of integrals( the second half of the book), so we'll discuss them again later.
x3 1 x2
y5
x
• Problem 7. Find the derivative of
e f x cos x 5
x3
f x 2x 1 x 2 x ln 8
f x log8 x 2 x
Example 10: Find the derivative of f x loge x
1 1 f x x ln e x
You can expect this result from Rules1 and 2 involving natural logs.
The Derivative of
ln x
• When you studied logs in the past, you probably concentrated on common logs (that is, those with a base of 10), and avoided natural logarithms( base e) as much as possible. Well, we have bad news for you: Most of what you'll see from now on involves natural logs. In fact, common logs almost never show up in calculus. But thta's okay. All you have to do is memorize a bunch of rules, and you'll be fine.
The Derivative of
a
x
• You should recall from your precalculus days that we can x ln a rewrite a x as e . Keep in mind that ln a is just a constant, which gives us the naxt rule:
dy e x ln a ln a a x ln a Rule No.7: If y a , then dx
x
Given the pattern of this chapter, you can guess what's coming: another rule that incorporates the Chain Rule.
3
f x e
x3
3
Example 6: Find the derivative of f x e tan x
f x sec2 x e tan x
Example 7: Find the second derivative of f x e x
2
f x 2 xe x
Example 1: Find thederivative of f x ln x3
3x 2 3 f x 3 x x
.
• You could have done this another way. If you recall your rules for logarithms:
f x 8
4 x5
x 8
4 x5
20 x ln 8
4
Example 13: Find the derivavtie of f x sin x
f x sin x cos x ln
Example 14: Find the derivavtie of
The Derivative of
e
x
• As you'll see in Rule No.3, the derivative of e x is probably the easiest thing that you'll ever have to do in calculus.
Rule No. 3: If
• Rule No.6: If y loga u , then dy 1 du
dx u ln a dx
Example 8: Find the derivative of
1 f x x ln 10
f x log10 x
Example 9: Find the derivative of
1 1 ln a x
This leads us to our next rule:
• Rule No.5: If y loga x , then dy 1
dx
x ln a
Once again, incorporating the Chain Rule gives us a more useful formula:
First, rewrite this as
f x x x
f x e x ln x
. Then take the derivatie:
x f x e x ln x ln x e x ln x ln x 1 x x ln x 1 x
• Problem 1. Find the derivative of y 3 ln 5 x 2 4 x
• Problem 2. Find the derivative of f x ln sin x 5
• Problem 3. Find the derivative of f x e
Rule No. 4: If
y eu
, then
dy u du e dx dx
Example 4: Find the derivative of f x e3 x
f x e3 x 3 3e3 x
Example 5: Find the derivative of
f x e x 3x 2 3x 2 e x
2
f x 2e 4 x e
x2
2 x2
The Derivative of
log a x
• This derivative is actually a little trickeir than the derivatie of a natural log. First, if you remember your logarithm rules about change of a base, we can rewrite log a x this way:
5 6
Example 3: Find the derivative of f x ln cos x
sin x tan x cos x
f x
• Finding the derivative of a natural logarithm is just a matter of following a simple formuls.
y ln u
dy 1 du , then dx u dx
Remember: u is a function of
x
, and
du dx
is it's derivative.
You'll see how simple this rule is after we try a few examples.
dy du u a ln a Rule No.8: If y a , then dx dx
u
Example 11: Find the derivavtie of f x 3x
f x 3x ln 3
Example 11: Find the derivavtie of
Would you have thought of that? Remember this trick. It might come in handy! Okay. Ready for some practice? Here are some more solved problems. Cover the solutions and get cracking.
3 x7 4 x2
• Problem 4. Find the derivative of f x log 4 tan x • Problem 5. Find the derivative of f x log8 • Problem 6. Find the derivative of
y ex
, then
dy ex dx
That's not a typo. The derivative is the same as the original function! Incorporating the Chain Rule, we get a good formula for finding the derivative:
y ln x
, then
dy 1 dx x
This rule has a corollary that incorporates the Chain Rule and is actually a more useful rule to memorize: • Rule No.2: If
ln x3 3 ln x
1 3 f x 3 x x
Therefore,
Example 2: Find the derivative of f x ln 5x 3x 6 .
5 18 x f x 5x 3x
ln x log a x ln a
Review the unit on Prerequisite Mathemactics if this leaves you scratching your head. Anyway, because ln a is a constant, we can take the derivative and we get:
Chapter 9 Exponential and Logarithmic Functions, Part One
AP Math Bella
As with trigonometric functions, you'll be expected to remember all of the logarithmic and exponential functions you've studied in the past. If you're not sure about any of this stuff. review the unit on Prerequisite Mathematics. Also, this is only part one of our treatment of exponents and logs. Much of what you need to know about these functions requires knowledge of integrals( the second half of the book), so we'll discuss them again later.
x3 1 x2
y5
x
• Problem 7. Find the derivative of
e f x cos x 5
x3
f x 2x 1 x 2 x ln 8
f x log8 x 2 x
Example 10: Find the derivative of f x loge x
1 1 f x x ln e x
You can expect this result from Rules1 and 2 involving natural logs.
The Derivative of
ln x
• When you studied logs in the past, you probably concentrated on common logs (that is, those with a base of 10), and avoided natural logarithms( base e) as much as possible. Well, we have bad news for you: Most of what you'll see from now on involves natural logs. In fact, common logs almost never show up in calculus. But thta's okay. All you have to do is memorize a bunch of rules, and you'll be fine.
The Derivative of
a
x
• You should recall from your precalculus days that we can x ln a rewrite a x as e . Keep in mind that ln a is just a constant, which gives us the naxt rule:
dy e x ln a ln a a x ln a Rule No.7: If y a , then dx
x
Given the pattern of this chapter, you can guess what's coming: another rule that incorporates the Chain Rule.
3
f x e
x3
3
Example 6: Find the derivative of f x e tan x
f x sec2 x e tan x
Example 7: Find the second derivative of f x e x
2
f x 2 xe x
Example 1: Find thederivative of f x ln x3
3x 2 3 f x 3 x x
.
• You could have done this another way. If you recall your rules for logarithms:
f x 8
4 x5
x 8
4 x5
20 x ln 8
4
Example 13: Find the derivavtie of f x sin x
f x sin x cos x ln
Example 14: Find the derivavtie of
The Derivative of
e
x
• As you'll see in Rule No.3, the derivative of e x is probably the easiest thing that you'll ever have to do in calculus.
Rule No. 3: If
• Rule No.6: If y loga u , then dy 1 du
dx u ln a dx
Example 8: Find the derivative of
1 f x x ln 10
f x log10 x
Example 9: Find the derivative of
1 1 ln a x
This leads us to our next rule:
• Rule No.5: If y loga x , then dy 1
dx
x ln a
Once again, incorporating the Chain Rule gives us a more useful formula:
First, rewrite this as
f x x x
f x e x ln x
. Then take the derivatie:
x f x e x ln x ln x e x ln x ln x 1 x x ln x 1 x
• Problem 1. Find the derivative of y 3 ln 5 x 2 4 x
• Problem 2. Find the derivative of f x ln sin x 5
• Problem 3. Find the derivative of f x e
Rule No. 4: If
y eu
, then
dy u du e dx dx
Example 4: Find the derivative of f x e3 x
f x e3 x 3 3e3 x
Example 5: Find the derivative of
f x e x 3x 2 3x 2 e x
2
f x 2e 4 x e
x2
2 x2
The Derivative of
log a x
• This derivative is actually a little trickeir than the derivatie of a natural log. First, if you remember your logarithm rules about change of a base, we can rewrite log a x this way:
5 6
Example 3: Find the derivative of f x ln cos x
sin x tan x cos x
f x
• Finding the derivative of a natural logarithm is just a matter of following a simple formuls.
y ln u
dy 1 du , then dx u dx
Remember: u is a function of
x
, and
du dx
is it's derivative.
You'll see how simple this rule is after we try a few examples.
dy du u a ln a Rule No.8: If y a , then dx dx
u
Example 11: Find the derivavtie of f x 3x
f x 3x ln 3
Example 11: Find the derivavtie of
Would you have thought of that? Remember this trick. It might come in handy! Okay. Ready for some practice? Here are some more solved problems. Cover the solutions and get cracking.
3 x7 4 x2
• Problem 4. Find the derivative of f x log 4 tan x • Problem 5. Find the derivative of f x log8 • Problem 6. Find the derivative of
y ex
, then
dy ex dx
That's not a typo. The derivative is the same as the original function! Incorporating the Chain Rule, we get a good formula for finding the derivative: