![Page 1: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/1.jpg)
Derivatives of Exponential and LogarithmFunctions
10/17/2011
![Page 2: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/2.jpg)
The Derivative of y = ex
Recall!ex is the unique exponential function whose slope at x = 0 is 1:
m=1
limh→0
e0+h − e0
h= lim
h→0
eh − 1
h= 1
![Page 3: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/3.jpg)
The Derivative of y = ex
Recall!ex is the unique exponential function whose slope at x = 0 is 1:
m=1
limh→0
e0+h − e0
h= lim
h→0
eh − 1
h= 1
![Page 4: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/4.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 5: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/5.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 6: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/6.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 7: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/7.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 8: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/8.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 9: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/9.jpg)
The Derivative of y = ex ...
limh→0
eh − 1
h= 1
d
dxex = lim
h→0
ex+h − ex
h
= limh→0
ex(eh − 1
)h
= ex limh→0
eh − 1
h
= ex ∗ 1
Sod
dxex = ex
![Page 10: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/10.jpg)
The Chain Rule
TheoremLet u be a function of x. Then
d
dxeu = eu
du
dx.
![Page 11: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/11.jpg)
Examples
Calculate...
1. ddx e17x
= 17e17x
2. ddx esin x
= cos(x)esin x
3. ddx e
√x2+x
= 2x+12√x2+x
e√x2+1
Notice, every time:
d
dxef (x) = f ′(x)ef (x)
![Page 12: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/12.jpg)
Examples
Calculate...
1. ddx e17x = 17e17x
2. ddx esin x = cos(x)esin x
3. ddx e
√x2+x = 2x+1
2√x2+x
e√x2+1
Notice, every time:
d
dxef (x) = f ′(x)ef (x)
![Page 13: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/13.jpg)
Examples
Calculate...
1. ddx e17x = 17e17x
2. ddx esin x = cos(x)esin x
3. ddx e
√x2+x = 2x+1
2√x2+x
e√x2+1
Notice, every time:
d
dxef (x) = f ′(x)ef (x)
![Page 14: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/14.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!
Remember:y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey
=1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 15: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/15.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey
=1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 16: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/16.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey
=1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 17: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/17.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey
=1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 18: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/18.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey=
1
e ln(x)
=1
x
ddx ln(x) = 1
x
![Page 19: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/19.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey=
1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 20: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/20.jpg)
The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!Remember:
y = ln x =⇒ ey = x
Take a derivative of both sides of ey = x to get
dy
dxey = 1
Sody
dx=
1
ey=
1
e ln(x)=
1
x
ddx ln(x) = 1
x
![Page 21: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/21.jpg)
Does it make sense?
ddx ln(x) = 1
x
f (x) = ln(x) 1 2 3 4
-1
1
f (x) = 1x
1 2 3 4
1
2
3
![Page 22: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/22.jpg)
Examples
Calculate
1. ddx ln x2
=2x
x2=
2
x
2. ddx ln(sin(x2))
=2x cos(x2)
sin(x2)
Notice, every time:
d
dxln(f (x)) =
f ′(x)
f (x)
![Page 23: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/23.jpg)
Examples
Calculate
1. ddx ln x2 =
2x
x2=
2
x
2. ddx ln(sin(x2)) =
2x cos(x2)
sin(x2)
Notice, every time:
d
dxln(f (x)) =
f ′(x)
f (x)
![Page 24: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/24.jpg)
Examples
Calculate
1. ddx ln x2 =
2x
x2=
2
x
2. ddx ln(sin(x2)) =
2x cos(x2)
sin(x2)
Notice, every time:
d
dxln(f (x)) =
f ′(x)
f (x)
![Page 25: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/25.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x
=d
dxex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 26: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/26.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x
=d
dxex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 27: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/27.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x =
d
dxex ln(2)
= ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 28: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/28.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x =
d
dxex ln(2) = ln(2) ∗ ex ln(2)
= ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 29: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/29.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x =
d
dxex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 30: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/30.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x =
d
dxex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x)
=1
ln(2) ∗ x
![Page 31: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/31.jpg)
The Calculus Standards: ex and ln x
To get the other derivatives:
ax = ex ln a
loga x =ln x
ln a
For example:
d
dx2x =
d
dxex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try:d
dxlog2(x) =
1
ln(2) ∗ x
![Page 32: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/32.jpg)
Differential equations
Suppose y is some mystery function of x and satisfies the equation
y ′ = ky
Goal: What is y??
1. If k = 1, then y = ex has this property and thus solves theequation.
2. For any k , y = ekx solves the equation too!
This equation, ddx y = ky is an example of a differential equation.
![Page 33: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/33.jpg)
Differential equations
Suppose y is some mystery function of x and satisfies the equation
y ′ = ky
Goal: What is y??
1. If k = 1, then y = ex has this property and thus solves theequation.
2. For any k , y = ekx solves the equation too!
This equation, ddx y = ky is an example of a differential equation.
![Page 34: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/34.jpg)
Differential equations
Suppose y is some mystery function of x and satisfies the equation
y ′ = ky
Goal: What is y??
1. If k = 1, then y = ex has this property and thus solves theequation.
2. For any k , y = ekx solves the equation too!
This equation, ddx y = ky is an example of a differential equation.
![Page 35: Derivatives of Exponential and Logarithm Functions · The Calculus Standards: ex and lnx To get the other derivatives: ax = ex lna log a x = lnx lna For example: d dx 2x = d dx ex](https://reader030.vdocuments.net/reader030/viewer/2022041106/5f079aec7e708231d41dd195/html5/thumbnails/35.jpg)
Differential equations
Suppose y is some mystery function of x and satisfies the equation
y ′ = ky
Goal: What is y??
1. If k = 1, then y = ex has this property and thus solves theequation.
2. For any k , y = ekx solves the equation too!
This equation, ddx y = ky is an example of a differential equation.