# Chain Rule

#### oregon88

I have this question and I dont know where to start.

The average commission c (in dollars) per trade by an investment corporation decreased as more of its customers traded online, according to the formula

$$\displaystyle c(u) = 100u^2 - 160u + 110$$ [units are dollars per trade] [u = fraction of trades done online]

During that time, the fraction of online trades increased according to

$$\displaystyle u(t) = 0.42 + 0.02t$$ [t = months since January 1, 1998]

Use direct substitution to express the average commission per trade c as a function of time t (do not just simplify the expression) and then use the chain rule to derive the formula for estimate the rate of change of average commission per trade. Use this formula to estimate the rate of change of average commission per trade at the beginning of October, 1998.

#### pickslides

MHF Helper
Find $$\displaystyle c'(u)$$ and $$\displaystyle u'(t)$$

Note that,

$$\displaystyle c(u) = 100u^2 - 160u + 110 \implies c(t) = 100(0.42 + 0.02t)^2 - 160(0.42 + 0.02t) + 110$$

$$\displaystyle c'(t) =c'(u) \times u'(t)$$

#### oregon88

ok so

$$\displaystyle c'(t) = (.02)(200u-160)$$
then I would use
$$\displaystyle c(t) = 100(0.42 + 0.02t)^2 - 160(0.42 + 0.02t) + 110$$
to find the rate of change of average comission per trade that would be at the beginning of October, 1998 which would be

$$\displaystyle c(10) = 100(0.42 + 0.02(10))^2 - 160(0.42 + 0.02(10)) + 110$$

does this sound right?

#### pickslides

MHF Helper
$$\displaystyle c(u) = 100u^2 - 160u + 110 \implies c(t)$$ $$\displaystyle = 100(0.42 + 0.02t)^2 - 160(0.42 + 0.02t) + 110 \implies c'(t)$$ $$\displaystyle = 100\times 2\times 0.02 \times (0.42 + 0.02t) - 160\times 0.02$$

#### oregon88

Ok so I think I finally have it.

$$\displaystyle c(u) = 100u^2-160u+110$$

and

$$\displaystyle u(t) = 0.42+0.02t$$

this would give you

$$\displaystyle c(t) = 100(0.42 + 0.02t)^2 - 160(0.42 + 0.02t) + 110$$
then $$\displaystyle c'(t) = 4(.42 + .02t)-3.2$$

my question is to find the rate of change of average commission per trade at the beginning of october, 1998 would you use c(9) or c(10) if t=months since January,1 1998?

#### drumist

my question is to find the rate of change of average commission per trade at the beginning of october, 1998 would you use c(9) or c(10) if t=months since January,1 1998?
October is the 10th month of the year, and January is the 1st month. So, October is 9 months after January.

#### oregon88

So the final answer is this?

$$\displaystyle c(9)=100(0.42+0.02(9)^2-160(0.42+0.02(9))+110 = -60$$

$$\displaystyle c'(-60)=4(0.42+0.02(-60)-3.2 = - 15.92$$

So the estimated rate of change of average commission per trade at the beginning of October, 1998 is = -15.92

This seems right but I have the funny feeling it isnt.

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