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    ShelleyMaeSilaganMartos
    The document provides solutions to 21 differentiation problems using the chain rule. It works through each problem step-by-step, applying the chain rule to functions with multiple nested layers and determining derivatives. The final three problems require a more formal use of the chain rule to determine derivatives, tangent lines, and functions from given properties.

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    P P P P P P P P PPPP PPPPPP PP PP PPP PPPP PP: C C CC

    Uploaded by

    ShelleyMaeSilaganMartos
    25 views16 pages
    The document provides solutions to 21 differentiation problems using the chain rule. It works through each problem step-by-step, applying the chain rule to functions with multiple nested layers and determining derivatives. The final three problems require a more formal use of the chain rule to determine derivatives, tangent lines, and functions from given properties.

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    Functions

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    The document provides solutions to 21 differentiation problems using the chain rule. It works through each problem step-by-step, applying the chain rule to functions with multiple nested layers and determining derivatives. The final three problems require a more formal use of the chain rule to determine derivatives, tangent lines, and functions from given properties.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    25 views16 pages

    P P P P P P P P PPPP PPPPPP PP PP PPP PPPP PP: C C CC

    Uploaded by

    ShelleyMaeSilaganMartos
    The document provides solutions to 21 differentiation problems using the chain rule. It works through each problem step-by-step, applying the chain rule to functions with multiple nested layers and determining derivatives. The final three problems require a more formal use of the chain rule to determine derivatives, tangent lines, and functions from given properties.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
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    You are on page 1of 16

    SOLUTIONS TO DIFFFERENTIATION OF FUNCTIONS USING THE CHAIN RULE

    SOLUTION 1 : Differentiate

    ( 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,

    = 2 (3x+1) (3) = 6 (3x+1) . Click HERE to return to the list of problems.

    SOLUTION 2 : Differentiate

    . . unchanged. Then

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

    Click HERE to return to the list of problems.

    SOLUTION 3 : Differentiate

    . . unchanged. Then

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

    Click HERE to return to the list of problems.

    SOLUTION 4 : Differentiate

    . .

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

    unchanged. Then

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

    . Click HERE to return to the list of problems.

    SOLUTION 5 : Differentiate

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

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

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

    . Click HERE to return to the list of problems.

    SOLUTION 6 : Differentiate

    ( 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,

    . Click HERE to return to the list of problems.

    SOLUTION 7 : Differentiate

    . . . ) Thus,

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

    . Click HERE to return to the list of problems.

    SOLUTION 8 : Differentiate

    . . Recall

    ( The outer layer is ``2 raised to a power'' and the inner layer is

    that . Differentiate ``2 raised to a power'' first, leaving unchanged. Then differentiate . ) Thus,

    . Click HERE to return to the list of problems.

    SOLUTION 9 : Differentiate

    ( Since 3 is a MULTIPLIED CONSTANT, we will first use the rule , 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,

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

    . Click HERE to return to the list of problems.

    SOLUTION 10 : Differentiate

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

    . Click HERE to return to the list of problems.

    SOLUTION 11 : Differentiate

    ( The outer layer is ``the common logarithm (base 10) function'' and the inner layer is . Recall that . Differentiate ``the common logarithm unchanged. Then differentiate .

    (base 10) function'' first, leaving ) Thus,

    . Click HERE to return to the list of problems.

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

    SOLUTION 12 : Differentiate

    ( Recall that , 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 ``the square'' first, leaving ``the cosine function'' and differentiate ``the cosine function'', leaving of . ) Thus, . Differentiate

    unchanged. Then

    unchanged. Finish with the derivative

    . Click HERE to return to the list of problems.

    SOLUTION 13 : Differentiate

    ( Since

    is a MULTIPLIED CONSTANT, we will first use the

    rule , where c is a constant. Hence, the constant just ``tags along'' during the differentiation process. It is NOT necessary to use the product rule. ) Thus,

    ( Recall that , 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 function'' and leaving . Differentiate ``the negative four power'' first, leaving ``the secant unchanged. Then differentiate ``the secant function'', unchanged. Finish with the derivative of .)

    . Click HERE to return to the list of problems.

    SOLUTION 14 : Differentiate

    ( 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 . Differentiate them in that order. ) Thus,

    . Click HERE to return to the list of problems.

    SOLUTION 15 : Differentiate

    ( 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,

    . Click HERE to return to the list of problems.

    SOLUTION 16 : Differentiate

    ( Since 10 is a MULTIPLIED CONSTANT, we will first use the rule , 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,

    ( 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 . Differentiate them in that order. )

    . Click HERE to return to the list of problems.

    SOLUTION 17 : Differentiate

    ( Since 4 is a MULTIPLIED CONSTANT, we will first use the rule , 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,

    ( 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 Differentiate them in that order. ) .

    . Click HERE to return to the list of problems.

    SOLUTION 18 : Differentiate

    ( 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,

    . The following three problems require a more formal use of the chain rule. Click HERE to return to the list of problems.

    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 . Thus,

    so that . Click HERE to return to the list of problems.

    SOLUTION 20 : Assume that If and of h at x=0 .

    , where f is a differentiable function.

    , determine an equation of the line tangent to the graph

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

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

    . Using the

    or

    Click HERE to return to the list of problems.

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

    Begin with

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

    iff Note that

    and , where C is any constant . Now think about ``reversing'' the process of differentiation. This is called finding an antiderivative. Thus,

    iff

    iff

    Since

    , we have

    so that

    and C = 2 . Thus,

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