# Math Help - Derivatives: Quotient Rule - Consumption Function

1. ## 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).

2. Originally Posted by shannon1111
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.

3. Thanks a lot, but now I am little confuse of the rule of derivative
Is there any shortcut way to remember the formula?

4. Originally Posted by shannon1111
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.

5. Thanks a lot, this works better for me!~