Find Dy/du Du/dx And Dy/dx

The Chain Dominion

The Chain Rule

Our goal is to differentiate functions such as

y = (3x + ane)¹⁰

The Chain Dominion

If
  y = y(u)

is a function of

u , and u = u(x)

is a part of

ten then

          dy           dy        du
   =
dx            du       dx

In our example nosotros have

        y  =  u ¹⁰

and

   u  =  3x + 1

so that

        dy/dx  =  (dy/du)(du/dx)

      =  (10u⁹) (3)  =  30u⁹  =  30 (3x+one)⁹

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Proof of the Chain Rule

Call up an alternate definition of the derivative:

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Examples

Find f '(ten) if

1. f(ten) = (10ⁱⁱⁱ - ten + 1)²⁰

2. f(10) = (x⁴ - 3x³ + ten)⁵

3. f(ten) = (one - x)⁹ (1-ten²)^four

4.                (10³ + 4x - three)⁷
   f(x)  =
   (2x - 1)ⁱⁱⁱ
   

Solution:

1. Here

     f(u) = u²⁰
   
   and

   u(x) = xⁱⁱⁱ - 10 + 1
   
   So that the derivative is

      [20u¹⁹] [3x² - 1]  =  [20(x³ - ten + ane)¹⁹] [3x² - 1]
   
   

2. Here

         f(u) = u⁵
   
   and

         u(ten) = x^four - 3xⁱⁱⁱ + ten
   
   So that the derivative is

           [5u⁴] [4x³ - 9xⁱⁱ + one]  =  [5(x⁴ - 3xⁱⁱⁱ + ten)⁴] [4x³ - 9x^two + i]
   
   

3. Here we need both the product and the chain dominion.

         f'(x) = [(1 - ten)⁹] [(1 - x²)^four]' + [(1 - x)⁹] '  [(1 - x^two)⁴]
   
   We first compute

       [(ane - x²)^iv] ' = [four(1 - x²)³] [-2x]
   
   and

           [(one - x)⁹] '  = [9(1 - x)^viii] [-1]
   
   Putting this all together gives

          f'(10) = [(1 - x)^ix] [4(i - x²)³] [-2x]  -  [9(1 - x)⁸]  [(i - x²)⁴]
   
   

4. Here we demand both the quotient and the chain rule.

   (2x - one)³ [(x³ + 4x - 3)⁷] '  -  (x³ + 4x - iii)⁷ [(2x - 1)^three] '
   f '(x) =
   (2x - 1)⁶
   
   We showtime compute

           [(x³ + 4x - 3)⁷] ' = [seven(ten^three + 4x - 3)^vi] [3x² + 4]
   
   and

      [(2x - 1)³] '  = [3(2x - 1)²] [2]
   
   Putting this all together gives

                        7(2x - 1)^three (x³ + 4x - three)⁶ (3x² + 4)  +  half dozen(ten^three + 4x - 3)^vii (2x - 1)²
   f '(x) =
   (2x - 1)⁶
   

   Exercise
   
   Find the derivative of

   10²(five - ten^three)^iv
   f(x)  =
   3 - x

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Application

Suppose that yous put $m into a banking company at an involvement charge per unit r compounded monthly for 3 years.  So the corporeality A that will exist in the business relationship at the end of the iii years will be

A = yard(1 + r/12)³⁶

Discover the charge per unit at which A rises with respect to a ascension in the interest rate when the interest rate is six%.

Solution

Nosotros are asked to find a derivative.  We use the chain rule with

        u  =  1 + r/12     and       A(u)  =  1000u³⁶

The 2 derivatives are

       u'  =  i/12      and     A'  =  36000u³⁵

The chain rule gives

        dA/dr  =  dA/du du/dr  =  (ane/12) 36000 u³⁵

        =  3000 u³⁵  =  3000(1 + r/12)³⁵

Now plug in r   =   6%  =  0.06  to get

        3000(1 + 0.06/12)³⁵  =  3572.18

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Find Dy/du Du/dx And Dy/dx,

Source: https://ltcconline.net/greenl/courses/115/differentiation/chain.htm

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