SOLUTIONS TO DIFFFERENTIATION OF FUNCTIONS USING THE CHAIN RULE



SOLUTION 1 : Differentiate tex2html_wrap_inline528 .

( The outer layer is ``the square'' and the inner layer is (3x+1) . Differentiate ``the square'' first, leaving (3x+1) unchanged. Then differentiate (3x+1). ) Thus,

tex2html_wrap_inline536

= 2 (3x+1) (3)

= 6 (3x+1) .

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SOLUTION 2 : Differentiate tex2html_wrap_inline542 .

( The outer layer is ``the square root'' and the inner layer is tex2html_wrap_inline544 . Differentiate ``the square root'' first, leaving tex2html_wrap_inline544 unchanged. Then differentiate tex2html_wrap_inline544. ) Thus,

tex2html_wrap_inline550

tex2html_wrap_inline552

tex2html_wrap_inline554

tex2html_wrap_inline556

tex2html_wrap_inline558

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SOLUTION 3 : Differentiate tex2html_wrap_inline560 .

( The outer layer is ``the 30th power'' and the inner layer is tex2html_wrap_inline562 . Differentiate ``the 30th power'' first, leaving tex2html_wrap_inline562 unchanged. Then differentiate tex2html_wrap_inline562. ) Thus,

tex2html_wrap_inline568

tex2html_wrap_inline570

tex2html_wrap_inline572

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SOLUTION 4 : Differentiate tex2html_wrap_inline574 .

( The outer layer is ``the one-third power'' and the inner layer is tex2html_wrap_inline576 . Differentiate ``the one-third power'' first, leaving tex2html_wrap_inline576 unchanged. Then differentiate tex2html_wrap_inline576. ) Thus,

tex2html_wrap_inline582

tex2html_wrap_inline584

(At this point, we will continue to simplify the expression, leaving the final answer with no negative exponents.)

tex2html_wrap_inline586

tex2html_wrap_inline588

tex2html_wrap_inline590

tex2html_wrap_inline592

tex2html_wrap_inline594

tex2html_wrap_inline596

tex2html_wrap_inline598

tex2html_wrap_inline600

tex2html_wrap_inline602

tex2html_wrap_inline604

tex2html_wrap_inline606

tex2html_wrap_inline608 .

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SOLUTION 5 : Differentiate tex2html_wrap_inline610 .

( First, begin by simplifying the expression before we differentiate it. ) Thus,

tex2html_wrap_inline612 tex2html_wrap_inline614

( The outer layer is ``the negative four-fifths power'' and the inner layer is tex2html_wrap_inline616 . Differentiate ``the negative four-fifths power'' first, leaving tex2html_wrap_inline616 unchanged. Then differentiate tex2html_wrap_inline616. )

tex2html_wrap_inline622

tex2html_wrap_inline624

(At this point, we will continue to simplify the expression, leaving the final answer with no negative exponents.)

tex2html_wrap_inline626

tex2html_wrap_inline628

tex2html_wrap_inline630

tex2html_wrap_inline632

tex2html_wrap_inline634

tex2html_wrap_inline636

tex2html_wrap_inline638

tex2html_wrap_inline640

tex2html_wrap_inline642

tex2html_wrap_inline644

tex2html_wrap_inline646 .

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SOLUTION 6 : Differentiate tex2html_wrap_inline648 .

( The outer layer is ``the sine function'' and the inner layer is (5x) . Differentiate ``the sine function'' first, leaving (5x) unchanged. Then differentiate (5x) . ) Thus,

tex2html_wrap_inline656

tex2html_wrap_inline658

tex2html_wrap_inline660 .

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SOLUTION 7 : Differentiate tex2html_wrap_inline662 .

( The outer layer is ``the exponential function'' and the inner layer is tex2html_wrap_inline664 . Recall that tex2html_wrap_inline666 . Differentiate ``the exponential function'' first, leaving tex2html_wrap_inline664 unchanged. Then differentiate tex2html_wrap_inline664. ) Thus,

tex2html_wrap_inline672

tex2html_wrap_inline674

tex2html_wrap_inline676 .

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SOLUTION 8 : Differentiate tex2html_wrap_inline678 .

( The outer layer is ``2 raised to a power'' and the inner layer is tex2html_wrap_inline680 . Recall that tex2html_wrap_inline682 . Differentiate ``2 raised to a power'' first, leaving tex2html_wrap_inline680 unchanged. Then differentiate tex2html_wrap_inline680. ) Thus,

tex2html_wrap_inline688

tex2html_wrap_inline690

tex2html_wrap_inline692 .

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SOLUTION 9 : Differentiate tex2html_wrap_inline694 .

( Since 3 is a MULTIPLIED CONSTANT, we will first use the rule tex2html_wrap_inline696, where c is a constant . Hence, the constant 3 just ``tags along'' during the differentiation process. It is NOT necessary to use the product rule. ) Thus,

tex2html_wrap_inline700

( Now the outer layer is ``the tangent function'' and the inner layer is tex2html_wrap_inline702 . Differentiate ``the tangent function'' first, leaving tex2html_wrap_inline702 unchanged. Then differentiate tex2html_wrap_inline702. )

tex2html_wrap_inline708

tex2html_wrap_inline710

tex2html_wrap_inline712

tex2html_wrap_inline714

tex2html_wrap_inline716 .

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SOLUTION 10 : Differentiate tex2html_wrap_inline718 .

( The outer layer is ``the natural logarithm (base e) function'' and the inner layer is ( 17-x ) . Recall that tex2html_wrap_inline722. Differentiate ``the natural logarithm function'' first, leaving ( 17-x ) unchanged. Then differentiate ( 17-x ). ) Thus,

tex2html_wrap_inline728

tex2html_wrap_inline730

tex2html_wrap_inline732 .

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SOLUTION 11 : Differentiate tex2html_wrap_inline734 .

( The outer layer is ``the common logarithm (base 10) function'' and the inner layer is tex2html_wrap_inline736 . Recall that tex2html_wrap_inline738 . Differentiate ``the common logarithm (base 10) function'' first, leaving tex2html_wrap_inline736 unchanged. Then differentiate tex2html_wrap_inline736. ) Thus,

tex2html_wrap_inline744

tex2html_wrap_inline746

tex2html_wrap_inline748 .

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Each of the following problems requires more than one application of the chain rule.




SOLUTION 12 : Differentiate tex2html_wrap_inline750 .

( Recall that tex2html_wrap_inline752, which makes ``the square'' the outer layer, NOT ``the cosine function''. In fact, this problem has three layers. The first layer is ``the square'', the second layer is ``the cosine function'', and the third layer is tex2html_wrap_inline754 . Differentiate ``the square'' first, leaving ``the cosine function'' and tex2html_wrap_inline754 unchanged. Then differentiate ``the cosine function'', leaving tex2html_wrap_inline754 unchanged. Finish with the derivative of tex2html_wrap_inline754. ) Thus,

tex2html_wrap_inline762

tex2html_wrap_inline764

tex2html_wrap_inline766

tex2html_wrap_inline768

tex2html_wrap_inline770 .

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SOLUTION 13 : Differentiate tex2html_wrap_inline772 .

( Since tex2html_wrap_inline774 is a MULTIPLIED CONSTANT, we will first use the rule tex2html_wrap_inline696, where c is a constant. Hence, the constant tex2html_wrap_inline774 just ``tags along'' during the differentiation process. It is NOT necessary to use the product rule. ) Thus,

tex2html_wrap_inline782

( Recall that tex2html_wrap_inline784, which makes ``the negative four power'' the outer layer, NOT ``the secant function''. In fact, this problem has three layers. The first layer is ``the negative four power'', the second layer is ``the secant function'', and the third layer is tex2html_wrap_inline786 . Differentiate ``the negative four power'' first, leaving ``the secant function'' and tex2html_wrap_inline786 unchanged. Then differentiate ``the secant function'', leaving tex2html_wrap_inline786 unchanged. Finish with the derivative of tex2html_wrap_inline786. )

tex2html_wrap_inline794

tex2html_wrap_inline796

tex2html_wrap_inline798

tex2html_wrap_inline800

tex2html_wrap_inline802 .

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SOLUTION 14 : Differentiate tex2html_wrap_inline804 .

( There are four layers in this problem. The first layer is ``the natural logarithm (base e) function'', the second layer is ``the fifth power'', the third layer is ``the cosine function'', and the fourth layer is tex2html_wrap_inline806 . Differentiate them in that order. ) Thus,

tex2html_wrap_inline808

tex2html_wrap_inline810

tex2html_wrap_inline812

tex2html_wrap_inline814

tex2html_wrap_inline816

tex2html_wrap_inline818

tex2html_wrap_inline820 .

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SOLUTION 15 : Differentiate tex2html_wrap_inline822 .

( There are four layers in this problem. The first layer is ``the square root function'', the second layer is ``the sine function'', the third layer is `` 7x plus the natural logarithm (base e) function'', and the fourth layer is (5x) . Differentiate them in that order. ) Thus,

tex2html_wrap_inline828

tex2html_wrap_inline830

tex2html_wrap_inline832

tex2html_wrap_inline834

tex2html_wrap_inline836

tex2html_wrap_inline838

tex2html_wrap_inline840

tex2html_wrap_inline842 .

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SOLUTION 16 : Differentiate tex2html_wrap_inline844 .

( Since 10 is a MULTIPLIED CONSTANT, we will first use the rule tex2html_wrap_inline696, where c is a constant. Hence, the constant 10 just ``tags along'' during the differentiation process. It is NOT necessary to use the product rule. ) Thus,

tex2html_wrap_inline850

( Now there are four layers in this problem. The first layer is ``the fifth power'', the second layer is ``1 plus the third power '', the third layer is ``2 minus the ninth power'', and the fourth layer is tex2html_wrap_inline852 . Differentiate them in that order. )

tex2html_wrap_inline854

tex2html_wrap_inline856

tex2html_wrap_inline858

tex2html_wrap_inline860

tex2html_wrap_inline862 .

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SOLUTION 17 : Differentiate tex2html_wrap_inline864 .

( Since 4 is a MULTIPLIED CONSTANT, we will first use the rule tex2html_wrap_inline696, where c is a constant. Hence, the constant 4 just ``tags along'' during the differentiation process. It is NOT necessary to use the product rule. ) Thus,

tex2html_wrap_inline870

( There are four layers in this problem. The first layer is ``the natural logarithm (base e) function'', the second layer is ``the natural logarithm (base e) function'', the third layer is ``the natural logarithm (base e) function'', and the fourth layer is tex2html_wrap_inline872 . Differentiate them in that order. )

tex2html_wrap_inline874

tex2html_wrap_inline876

tex2html_wrap_inline878

tex2html_wrap_inline880

tex2html_wrap_inline882 .

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SOLUTION 18 : Differentiate tex2html_wrap_inline884 .

( There are four layers in this problem. The first layer is ``the third power'', the second layer is ``the tangent function'', the third layer is ``the square root function'', the fourth layer is ``the cotangent function'', and the fifth layer is (7x) . Differentiate them in that order. ) Thus,

tex2html_wrap_inline888

tex2html_wrap_inline890

tex2html_wrap_inline892

tex2html_wrap_inline894

tex2html_wrap_inline896

tex2html_wrap_inline898

tex2html_wrap_inline900 .


The following three problems require a more formal use of the chain rule.

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SOLUTION 19 : Assume that h(x) = f( g(x) ) , where both f and g are differentiable functions. If g(-1)=2, g'(-1)=3, and f'(2)=-4 , what is the value of h'(-1) ?

Recall that the chain rule states that tex2html_wrap_inline914 . Thus,

tex2html_wrap_inline916

so that

tex2html_wrap_inline918 .

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SOLUTION 20 : Assume that tex2html_wrap_inline920, where f is a differentiable function. If tex2html_wrap_inline924 and tex2html_wrap_inline926 , determine an equation of the line tangent to the graph of h at x=0 .

The outer layer of this function is ``the third power'' and the inner layer is f(x) . The chain rule gives us that the derivative of h is

tex2html_wrap_inline936 .

Thus, the slope of the line tangent to the graph of h at x=0 is

tex2html_wrap_inline942 .

This line passes through the point tex2html_wrap_inline944 . Using the point-slope form of a line, an equation of this tangent line is

tex2html_wrap_inline946 or tex2html_wrap_inline948 .

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SOLUTION 21 : Determine a differentiable function y = f(x) which has the properties tex2html_wrap_inline952 and tex2html_wrap_inline924.

Begin with tex2html_wrap_inline952 and assume that f(x) is not identically zero. Then

tex2html_wrap_inline952 iff tex2html_wrap_inline962 .

Note that

tex2html_wrap_inline964

and

tex2html_wrap_inline966, where C is any constant .

Now think about ``reversing'' the process of differentiation. This is called finding an antiderivative. Thus,

tex2html_wrap_inline962 iff tex2html_wrap_inline972

iff tex2html_wrap_inline974 .

Since tex2html_wrap_inline924, we have tex2html_wrap_inline978 so that tex2html_wrap_inline980 and C = 2 . Thus,

tex2html_wrap_inline984 .

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Duane Kouba
Fri May 9 12:13:55 PDT 1997