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Taking a derivative using leibniz’s notation, example 1

Find the derivative of y = ( x 3 x + 2 ) 5 .

First, let u = x 3 x + 2 . Thus, y = u 5 . Next, find d u d x and d y d u . Using the quotient rule,

d u d x = 2 ( 3 x + 2 ) 2

and

d y d u = 5 u 4 .

Finally, we put it all together.

d y d x = d y d u · d u d x Apply the chain rule. = 5 u 4 · 2 ( 3 x + 2 ) 2 Substitute d y d u = 5 u 4 and d u d x = 2 ( 3 x + 2 ) 2 . = 5 ( x 3 x + 2 ) 4 · 2 ( 3 x + 2 ) 2 Substitute u = x 3 x + 2 . = 10 x 4 ( 3 x + 2 ) 6 Simplify.

It is important to remember that, when using the Leibniz form of the chain rule, the final answer must be expressed entirely in terms of the original variable given in the problem.

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Taking a derivative using leibniz’s notation, example 2

Find the derivative of y = tan ( 4 x 2 3 x + 1 ) .

First, let u = 4 x 2 3 x + 1 . Then y = tan u . Next, find d u d x and d y d u :

d u d x = 8 x 3 and d y d u = sec 2 u .

Finally, we put it all together.

d y d x = d y d u · d u d x Apply the chain rule. = sec 2 u · ( 8 x 3 ) Use d u d x = 8 x 3 and d y d u = sec 2 u . = sec 2 ( 4 x 2 3 x + 1 ) · ( 8 x 3 ) Substitute u = 4 x 2 3 x + 1 .
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Use Leibniz’s notation to find the derivative of y = cos ( x 3 ) . Make sure that the final answer is expressed entirely in terms of the variable x .

d y d x = −3 x 2 sin ( x 3 )

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Key concepts

  • The chain rule allows us to differentiate compositions of two or more functions. It states that for h ( x ) = f ( g ( x ) ) ,
    h ( x ) = f ( g ( x ) ) g ( x ) .

    In Leibniz’s notation this rule takes the form
    d y d x = d y d u · d u d x .
  • We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them.
  • The chain rule combines with the power rule to form a new rule:
    If h ( x ) = ( g ( x ) ) n , then h ( x ) = n ( g ( x ) ) n 1 g ( x ) .
  • When applied to the composition of three functions, the chain rule can be expressed as follows: If h ( x ) = f ( g ( k ( x ) ) ) , then h ( x ) = f ( g ( k ( x ) ) g ( k ( x ) ) k ( x ) .

Key equations

  • The chain rule
    h ( x ) = f ( g ( x ) ) g ( x )
  • The power rule for functions
    h ( x ) = n ( g ( x ) ) n 1 g ( x )

For the following exercises, given y = f ( u ) and u = g ( x ) , find d y d x by using Leibniz’s notation for the chain rule: d y d x = d y d u d u d x .

y = 3 u 6 , u = 2 x 2

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y = 6 u 3 , u = 7 x 4

18 u 2 · 7 = 18 ( 7 x 4 ) 2 · 7

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y = sin u , u = 5 x 1

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y = cos u , u = x 8

sin u · −1 8 = sin ( x 8 ) · −1 8

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y = 4 u + 3 , u = x 2 6 x

8 x 24 2 4 u + 3 = 4 x 12 4 x 2 24 x + 3

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For each of the following exercises,

  1. decompose each function in the form y = f ( u ) and u = g ( x ) , and
  2. find d y d x as a function of x .

y = ( 3 x 2 + 1 ) 3

a. u = 3 x 2 + 1 ; b. 18 x ( 3 x 2 + 1 ) 2

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y = ( x 7 + 7 x ) 7

a. f ( u ) = u 7 , u = x 7 + 7 x ; b. 7 ( x 7 + 7 x ) 6 · ( 1 7 7 x 2 )

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y = csc ( π x + 1 )

a. f ( u ) = csc u , u = π x + 1 ; b. π csc ( π x + 1 ) · cot ( π x + 1 )

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y = −6 sin −3 x

a. f ( u ) = −6 u −3 , u = sin x , b. 18 sin −4 x · cos x

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For the following exercises, find d y d x for each function.

y = ( 3 x 2 + 3 x 1 ) 4

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y = ( 5 2 x ) −2

4 ( 5 2 x ) 3

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y = ( 2 x 3 x 2 + 6 x + 1 ) 3

6 ( 2 x 3 x 2 + 6 x + 1 ) 2 ( 3 x 2 x + 3 )

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y = ( tan x + sin x ) −3

−3 ( tan x + sin x ) −4 · ( sec 2 x + cos x )

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y = sin ( cos 7 x )

−7 cos ( cos 7 x ) · sin 7 x

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y = cot 3 ( 4 x + 1 )

−12 cot 2 ( 4 x + 1 ) · csc 2 ( 4 x + 1 )

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Let y = [ f ( x ) ] 3 and suppose that f ( 1 ) = 4 and d y d x = 10 for x = 1 . Find f ( 1 ) .

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Let y = ( f ( x ) + 5 x 2 ) 4 and suppose that f ( −1 ) = −4 and d y d x = 3 when x = −1 . Find f ( −1 )

10 3 4

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Let y = ( f ( u ) + 3 x ) 2 and u = x 3 2 x . If f ( 4 ) = 6 and d y d x = 18 when x = 2 , find f ( 4 ) .

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[T] Find the equation of the tangent line to y = sin ( x 2 ) at the origin. Use a calculator to graph the function and the tangent line together.

y = −1 2 x

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[T] Find the equation of the tangent line to y = ( 3 x + 1 x ) 2 at the point ( 1 , 16 ) . Use a calculator to graph the function and the tangent line together.

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Find the x -coordinates at which the tangent line to y = ( x 6 x ) 8 is horizontal.

x = ± 6

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[T] Find an equation of the line that is normal to g ( θ ) = sin 2 ( π θ ) at the point ( 1 4 , 1 2 ) . Use a calculator to graph the function and the normal line together.

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For the following exercises, use the information in the following table to find h ( a ) at the given value for a .

x f ( x ) f ( x ) g ( x ) g ( x )
0 2 5 0 2
1 1 −2 3 0
2 4 4 1 −1
3 3 −3 2 3

h ( x ) = f ( g ( x ) ) ; a = 0

10

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h ( x ) = g ( f ( x ) ) ; a = 0

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h ( x ) = ( x 4 + g ( x ) ) −2 ; a = 1

1 8

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h ( x ) = ( f ( x ) g ( x ) ) 2 ; a = 3

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h ( x ) = f ( x + f ( x ) ) ; a = 1

−4

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h ( x ) = ( 1 + g ( x ) ) 3 ; a = 2

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h ( x ) = g ( 2 + f ( x 2 ) ) ; a = 1

−12

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h ( x ) = f ( g ( sin x ) ) ; a = 0

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[T] The position function of a freight train is given by s ( t ) = 100 ( t + 1 ) −2 , with s in meters and t in seconds. At time t = 6 s, find the train’s

  1. velocity and
  2. acceleration.
  3. Using a. and b. is the train speeding up or slowing down?

a. 200 343 m/s, b. 600 2401 m/s 2 , c. The train is slowing down since velocity and acceleration have opposite signs.

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[T] A mass hanging from a vertical spring is in simple harmonic motion as given by the following position function, where t is measured in seconds and s is in inches:

s ( t ) = −3 cos ( π t + π 4 ) .

  1. Determine the position of the spring at t = 1.5 s.
  2. Find the velocity of the spring at t = 1.5 s.
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[T] The total cost to produce x boxes of Thin Mint Girl Scout cookies is C dollars, where C = 0.0001 x 3 0.02 x 2 + 3 x + 300 . In t weeks production is estimated to be x = 1600 + 100 t boxes.

  1. Find the marginal cost C ( x ) .
  2. Use Leibniz’s notation for the chain rule, d C d t = d C d x · d x d t , to find the rate with respect to time t that the cost is changing.
  3. Use b. to determine how fast costs are increasing when t = 2 weeks. Include units with the answer.

a. C ( x ) = 0.0003 x 2 0.04 x + 3 b. d C d t = 100 · ( 0.0003 x 2 0.04 x + 3 ) c. Approximately $90,300 per week

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[T] The formula for the area of a circle is A = π r 2 , where r is the radius of the circle. Suppose a circle is expanding, meaning that both the area A and the radius r (in inches) are expanding.

  1. Suppose r = 2 100 ( t + 7 ) 2 where t is time in seconds. Use the chain rule d A d t = d A d r · d r d t to find the rate at which the area is expanding.
  2. Use a. to find the rate at which the area is expanding at t = 4 s.
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[T] The formula for the volume of a sphere is S = 4 3 π r 3 , where r (in feet) is the radius of the sphere. Suppose a spherical snowball is melting in the sun.

  1. Suppose r = 1 ( t + 1 ) 2 1 12 where t is time in minutes. Use the chain rule d S d t = d S d r · d r d t to find the rate at which the snowball is melting.
  2. Use a. to find the rate at which the volume is changing at t = 1 min.

a. d S d t = 8 π r 2 ( t + 1 ) 3 b. The volume is decreasing at a rate of π 36 ft 3 /min.

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[T] The daily temperature in degrees Fahrenheit of Phoenix in the summer can be modeled by the function T ( x ) = 94 10 cos [ π 12 ( x 2 ) ] , where x is hours after midnight. Find the rate at which the temperature is changing at 4 p.m.

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[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. The depth is modeled by the function D ( t ) = 5 sin ( π 6 t 7 π 6 ) + 8 , where t is the number of hours after midnight. Find the rate at which the depth is changing at 6 a.m.

~ 2.3 ft/hr

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Questions & Answers

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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