How to arrive at this answer?

[reardear]

A sum of 1400 dollars is deposited in an account with an interest rate of r percent per year, compunded daily. At the end of 6 years, the balance in the account is given by
\(\displaystyle A=1400\left( 1+\dfrac {r} {36500}\right) ^{2190}\)
Find the rate of change of A with respect to r if the interest rate is r%.


I know that the following is the answer, and it looks like the power rule was used, but that isn't the case. How the heck did it end up at 84, and nothing inside the brackets changed? I also don't understand what the question is asking (find rate of change of A with respect to r). Any help would be greatly appreciated as always!
\(\displaystyle A=84\left( 1+\dfrac {r} {36500}\right) ^{2189}\)

 

[tkhunny]

Given \(\displaystyle A(r) = 1400\cdot \left(1 + \frac{r}{36500}\right)^{2190}\),

We have \(\displaystyle \frac{d}{dr}A(r) = 1400\cdot 2190\cdot\left(1 + \frac{r}{36500}\right)^{2189}\cdot\frac{1}{36500}\)

Unfortunately, you did not show any work at all. This makes it quite difficult to see where you wandered off.

 

[reardear]

Given \(\displaystyle A(r) = 1400\cdot \left(1 + \frac{r}{36500}\right)^{2190}\),

We have \(\displaystyle \frac{d}{dr}A(r) = 1400\cdot 2190\cdot\left(1 + \frac{r}{36500}\right)^{2189}\cdot\frac{1}{36500}\)

Unfortunately, you did not show any work at all. This makes it quite difficult to see where you wandered off.

\(\displaystyle A(r) = 3066000\cdot \left(1 + \frac{r}{36500}\right)^{2189}\)
I got to there and didn't know what else to do. I didn't know what to do about the r. I forgot I could just use the power rule and get \(\displaystyle \frac{1}{36500}\). Thanks a lot!

 

[JeffM]

\(\displaystyle A(r) = 3066000\cdot \left(1 + \frac{r}{36500}\right)^{2189}\)
I got to there and didn't know what else to do. I didn't know what to do about the r. I forgot I could just use the power rule and get \(\displaystyle \frac{1}{36500}\). Thanks a lot!

You used the power rule to get to \(\displaystyle 2190 * 1400 * \left(1 + \dfrac{r}{36500}\right)^{2190 - 1} = 3,066,000\left(1 + \dfrac{r}{36500}\right)^{2189}.\)

You forgot to use the chain rule.