Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.
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Transcript of Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.
![Page 1: Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f0e5503460f94c22209/html5/thumbnails/1.jpg)
Product & quotient rules & higher-order
derivatives (2.3) October 17th, 2012
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I. the product rule
Thm. 2.7: The Product Rule: The product of two differentiable functions f and g is differentiable. The derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.
d
dx[ f (x)g(x)]= f(x)g'(x)+ g(x) f '(x)
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Ex.1: Find the derivative of each function.
(a)h(x)=(5x2 −3x)(4 + 6x)
(b)y=4x3 cosx
(c)y=4xsinx−4cosx
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You Try: Find the derivative of each function.
(a) f (x)=(9x−2)(4x2 +1)
(b)g(s)= s(4 −s2 )
(c)g(x)= xsinx
![Page 5: Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f0e5503460f94c22209/html5/thumbnails/5.jpg)
II. the quotient rule
Thm. 2.8: The Quotient Rule: The quotient f/g of two differentiable functions f and g is differentiable for all value of x for which .
g(x)≠0
d
dx
f (x)
g(x)
⎡
⎣⎢
⎤
⎦⎥=
g(x) f '(x)− f(x)g'(x)[g(x)]2
,g(x) ≠0
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A. using the quotient rule
Ex. 2: Find the derivative of .y=4x+52x2 −1
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You Try: Find the derivative of .f (x)=3x2 −x2x+5
![Page 8: Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f0e5503460f94c22209/html5/thumbnails/8.jpg)
B. Rewriting before differentiating
Ex. 3:Find the slope of the tangent line to the graph of at (-1, -7/3).
f (x)=4 −3 / xx−2
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You Try: Find an equation of the tangent line to the graph of at (1, 3).
y=1+2 / x4x−3
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*If it is unnecessary to differentiate a function by the quotient rule, it is better to use the constant multiple rule.
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c. using the constant multiple rule
Ex. 4: Find the derivative of each function.
(a)
(b)
y=3x2 −x
2
f (x)=2x3 −4x
3x
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You Try: Find the derivative of each function.
(a)
(b)
y=4
3x3
g(x)=−2(4x2 −x)
3x
![Page 13: Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f0e5503460f94c22209/html5/thumbnails/13.jpg)
III. derivatives of trigonometric functions
Thm. 2.9: Derivatives of Trigonometric Functions:
d
dx[tan x]=sec2 x
d
dx[cot x]=−csc2 x
d
dx[sec x]=secxtanx
d
dx[csc x]=−cscxcotx
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A. Proof of the derivative of sec x
Ex. 5: Prove .d
dx[sec x]=secxtanx
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B. differentiating trigonometric functions
Ex. 6: Find the derivative of each function.
(a)
(b)
y=−2x3 −cotx
y=xcscx
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C. Different forms of a derivative
Ex. 7:Differentiate both forms of .
y=1+sinxcosx
=secx+ tanx
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IV. Higher-order derivatives*We know that we differentiate the position function of an object to obtain the velocity function. We also differentiate the velocity function to obtain the acceleration function. Or, you could differentiate the position function twice to obtain the acceleration function.
s(t) position function
v(t) = s’(t) velocity function
a(t) = v’(t) = s’’(t) acceleration function
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*Higher-order derivatives are denoted as follows:
First derivative
Second derivative
Third derivative
Fourth derivative . . .nth derivative
y ', f '(x),dy
dx,d
dx[ f (x)],Dx[y]
y '', f ''(x),d 2y
dx2,d 2
dx2[ f (x)],Dx
2[y]
y ''', f '''(x),d 3y
dx3,d 3
dx3[ f (x)],Dx
3[y]
y(4 ), f (4 )(x),d 4y
dx4,d 4
dx4[ f (x)],Dx
4[y]
y(n), f (n)(x),d ny
dxn,d n
dxn[ f (x)],Dx
n[y]
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A. finding acceleration
Ex. 8: Given the position function , find the acceleration at 5 seconds. (Let s(t) be in feet).
s(t)=t+ 32t−2
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You Try: Given the position function , where s(t) is in
feet, find that acceleration at 10 seconds.
s(t)=t2 +2t−1
t