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Math Help - Derivatives: Quotient Rule - Consumption Function

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    Derivatives: Quotient Rule - Consumption Function

    The consumption function is given by


    c=((14squar root of I^3)-35)/(I+9)


    where I is the total national income and C the total national consumption, both expressed in billions of dollars.

    A)The marginal propensity to consume is defined as the rate of change of consumption with respect to income:
    Marginal propensity to consume = dC/dI

    The marginal propensity to consume for the consumption function C given above when I=81 equals ______(billions of dollars)/(one billion of dollars of change of the income).




    B)The total national savings is defined as the difference between income I and consumption C:
    S=I-C

    The marginal propensity to save is defined as the rate of change of savings with respect to income:

    Marginal propensity to save =dS/dI=1-dC/dI

    The marginal propensity to save for the consumption function C given above when I=81 equals ____(billions of dollars)/(one billion of dollars of change of the income).
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    Quote Originally Posted by shannon1111 View Post
    The consumption function is given by


    c=((14squar root of I^3)-35)/(I+9)


    where I is the total national income and C the total national consumption, both expressed in billions of dollars.

    A)The marginal propensity to consume is defined as the rate of change of consumption with respect to income:
    Marginal propensity to consume = dC/dI

    The marginal propensity to consume for the consumption function C given above when I=81 equals ______(billions of dollars)/(one billion of dollars of change of the income).
    We want \frac{dC}{dI}, where C = \frac{14\sqrt{I^3}-35}{I+9}, right?

    So, we differentiate, using the quotient rule:

    \frac d{dI}\left[\frac{14\sqrt{I^3}-35}{I+9}\right]

    =\frac d{dI}\left[\frac{14I^{3/2}-35}{I+9}\right]

    =\frac{(I+9)\left(21I^{1/2}\right) - \left(14I^{3/2} - 35\right)}{(I+9)^2}

    Clean it up, and then set I equal to 81. Then substitute this value into the expression for part B.
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    Thanks a lot, but now I am little confuse of the rule of derivative
    Is there any shortcut way to remember the formula?
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    Quote Originally Posted by shannon1111 View Post
    Thanks a lot, but now I am little confuse of the rule of derivative
    Is there any shortcut way to remember the formula?
    The quotient rule is \left(\frac uv\right)' = \frac{vu' - uv'}{v^2}\text{.} The only tricky part is remembering which of the terms in the numerator comes first.

    For that, my first Calc teacher always started differentiating a quotient by exclaiming "bottoms up!" and pretending to take a swig of beer. "Bottoms up" means you begin by bringing the bottom of the fraction ( v) up to the top (without differentiating); then you multiply by the derivative of the other one ( u'), and like the product rule, you then do the reverse uv', only subtracting rather than adding. Then remember to square the original denominator.

    That's the best I can do. If someone has a good mnemonic, I'd love to hear it.
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    Thanks a lot, this works better for me!~
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