Derivatives of Products and Quotients

Сентябрь 13, 2022

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Derivatives of Products and Quotients

Содержание

2. Barnett/Ziegler/Byleen Business Calculus 11e Derivatives of Products In words: The derivative of the product of two
3. Barnett/Ziegler/Byleen Business Calculus 11e Example Find the derivative of y = 5x2(x3 + 2).
4. Barnett/Ziegler/Byleen Business Calculus 11e Example Find the derivative of y = 5x2(x3 + 2). Solution: Let
5. Barnett/Ziegler/Byleen Business Calculus 11e Theorem 2 (Quotient Rule) If f (x) = T (x) / B(x),
6. Barnett/Ziegler/Byleen Business Calculus 11e Example Find the derivative of y = 3x / (2x + 5).
7. Barnett/Ziegler/Byleen Business Calculus 11e Example Find the derivative of y = 3x / (2x + 5).
8. Barnett/Ziegler/Byleen Business Calculus 11e Tangent Lines Let f (x) = (2x - 9)(x2 + 6). Find
9. Barnett/Ziegler/Byleen Business Calculus 11e Tangent Lines Let f (x) = (2x - 9)(x2 + 6). Find
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Слайд 2

Barnett/Ziegler/Byleen Business Calculus 11e
Derivatives of Products
In words: The derivative of the

product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Theorem 1 (Product Rule)
If f (x) = F(x) ⋅ S(x), and if F ’(x) and S ’(x) exist, then
f ’ (x) = F(x) ⋅ S ’(x) + F ’(x) ⋅ S(x), or

Слайд 3

Barnett/Ziegler/Byleen Business Calculus 11e
Example
Find the derivative of y = 5x2(x3 +

2).

Слайд 4

Barnett/Ziegler/Byleen Business Calculus 11e
Example
Find the derivative of y = 5x2(x3 +

2).
Solution:
Let F(x) = 5x2, so F ’(x) = 10x Let S(x) = x3 + 2, so S ’(x) = 3x2.
Then
f ’ (x) = F(x) ⋅ S ’(x) + F ’(x) ⋅ S(x)
= 5x2 ⋅ 3x2 + 10x ⋅ (x3 + 2)
= 15x4 + 10x4 + 20x = 25x4 + 20x.

Слайд 5

Barnett/Ziegler/Byleen Business Calculus 11e
Theorem 2 (Quotient Rule)
If f (x) = T

(x) / B(x), and if T ’(x) and B ’(x) exist, then
or

Derivatives of Quotients

In words: The derivative of the quotient of two functions is the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all over the bottom function squared.

Слайд 6

Barnett/Ziegler/Byleen Business Calculus 11e
Example
Find the derivative of y = 3x /

(2x + 5).

Слайд 7

Barnett/Ziegler/Byleen Business Calculus 11e
Example
Find the derivative of y = 3x /

(2x + 5).
Solution:
Let T(x) = 3x, so T ’(x) = 3 Let B(x) = 2x + 5, so B ’(x) = 2.
Then

Слайд 8

Barnett/Ziegler/Byleen Business Calculus 11e
Tangent Lines
Let f (x) = (2x - 9)(x2

+ 6). Find the equation of the line tangent to the graph of f (x) at x = 3.

Слайд 9

Barnett/Ziegler/Byleen Business Calculus 11e
Tangent Lines
Let f (x) = (2x - 9)(x2

+ 6). Find the equation of the line tangent to the graph of f (x) at x = 3.
Solution: First, find f ’(x):
f ’(x) = (2x - 9) (2x) + (2) (x2 + 6)
Then find f (3) and f ’(3):
f (3) = -45 f ’(3) = 12
The tangent has slope 12 and goes through the point (3, -45).
Using the point-slope form y - y1 = m(x - x1), we get
y – (-45) = 12(x - 3) or y = 12x - 81

Derivatives of Products and Quotients

Содержание

Barnett/Ziegler/Byleen Business Calculus 11eDerivatives of ProductsIn words: The derivative of the

Barnett/Ziegler/Byleen Business Calculus 11eExampleFind the derivative of y = 5x2(x3 +

Barnett/Ziegler/Byleen Business Calculus 11eExampleFind the derivative of y = 5x2(x3 +

Barnett/Ziegler/Byleen Business Calculus 11eTheorem 2 (Quotient Rule)If f (x) = T

Barnett/Ziegler/Byleen Business Calculus 11eExampleFind the derivative of y = 3x /

Barnett/Ziegler/Byleen Business Calculus 11eExampleFind the derivative of y = 3x /

Barnett/Ziegler/Byleen Business Calculus 11eTangent LinesLet f (x) = (2x - 9)(x2

Barnett/Ziegler/Byleen Business Calculus 11eTangent LinesLet f (x) = (2x - 9)(x2

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