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Derivatives of Exponential and Logarithm Functions 10/17/2011 The - - PowerPoint PPT Presentation
Derivatives of Exponential and Logarithm Functions 10/17/2011 The - - PowerPoint PPT Presentation
Derivatives of Exponential and Logarithm Functions 10/17/2011 The Derivative of y = e x Recall! e x is the unique exponential function whose slope at x = 0 is 1: m=1 The Derivative of y = e x Recall! e x is the unique exponential function whose
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The Derivative of y = ex
Recall! ex is the unique exponential function whose slope at x = 0 is 1:
m=1
lim
h→0
e0+h − e0 h = lim
h→0
eh − 1 h = 1
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1 d dx ex = lim
h→0
ex+h − ex h
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1 d dx ex = lim
h→0
ex+h − ex h = lim
h→0
ex eh − 1
- h
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1 d dx ex = lim
h→0
ex+h − ex h = lim
h→0
ex eh − 1
- h
= ex lim
h→0
eh − 1 h
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1 d dx ex = lim
h→0
ex+h − ex h = lim
h→0
ex eh − 1
- h
= ex lim
h→0
eh − 1 h = ex ∗ 1
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The Derivative of y = ex...
lim
h→0
eh − 1 h = 1 d dx ex = lim
h→0
ex+h − ex h = lim
h→0
ex eh − 1
- h
= ex lim
h→0
eh − 1 h = ex ∗ 1 So d dx ex = ex
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The Chain Rule
Theorem
Let u be a function of x. Then d dx eu = eu du dx .
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Examples
Calculate... 1.
d dx e17x
2.
d dx esin x
3.
d dx e √ x2+x
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Examples
Calculate... 1.
d dx e17x = 17e17x
2.
d dx esin x = cos(x)esin x
3.
d dx e √ x2+x = 2x+1 2 √ x2+x e √ x2+1
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Examples
Calculate... 1.
d dx e17x = 17e17x
2.
d dx esin x = cos(x)esin x
3.
d dx e √ x2+x = 2x+1 2 √ x2+x e √ x2+1
Notice, every time: d dx ef (x) = f ′(x)ef (x)
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The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation!
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The Derivative of y = ln x
To find the derivative of ln(x), use implicit differentiation! Remember: y = ln x = ⇒ ey = x
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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 dx ey = 1
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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 dx ey = 1 So dy dx = 1 ey
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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 dx ey = 1 So dy dx = 1 ey = 1 eln(x)
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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 dx ey = 1 So dy dx = 1 ey = 1 eln(x) = 1 x
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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 dx ey = 1 So dy dx = 1 ey = 1 eln(x) = 1 x
d dx ln(x) = 1 x
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Does it make sense?
d dx ln(x) = 1 x
f (x) = ln(x)
1 2 3 4
- 1
1
f (x) = 1
x
1 2 3 4 1 2 3
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Examples
Calculate 1.
d dx ln x2
2.
d dx ln(sin(x2))
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Examples
Calculate 1.
d dx ln x2 = 2x
x2 = 2 x 2.
d dx ln(sin(x2)) = 2x cos(x2)
sin(x2)
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Examples
Calculate 1.
d dx ln x2 = 2x
x2 = 2 x 2.
d dx ln(sin(x2)) = 2x cos(x2)
sin(x2) Notice, every time: d dx ln(f (x)) = f ′(x) f (x)
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The Calculus Standards: ex and ln x
To get the other derivatives: ax = ex ln a loga x = ln x ln a
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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 dx 2x
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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 dx 2x = d dx ex ln(2)
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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 dx 2x = d dx ex ln(2) = ln(2) ∗ ex ln(2)
(ln(2) is a constant!!!)
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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 dx 2x = d dx ex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
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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 dx 2x = d dx ex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try: d dx log2(x)
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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 dx 2x = d dx ex ln(2) = ln(2) ∗ ex ln(2) = ln(2) ∗ 2x
(ln(2) is a constant!!!)
You try: d dx log2(x) = 1 ln(2) ∗ x
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Differential equations
Suppose y is some mystery function of x and satisfies the equation y′ = ky Goal: What is y??
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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 the
equation.
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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 the
equation.
- 2. For any k, y = ekx solves the equation too!
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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 the
equation.
- 2. For any k, y = ekx solves the equation too!