calculus interest rate question

Question

If 1300 dollars is invested at an annual interest rate r  compounded monthly, the amount in the account at the end of 4 years is given by

A = 1300 (1 + 1/12(r) ) ^48

Find the rate of change of the amount A with respect to the rate r for the following values of r:

r = 4.5 percent: 

r  = 6.5 percent: 

Answer 1

Constant rule: d c / d x = 0 where c is a real number

Power rule of differentiation: d xⁿ /dx = n x^n-1

Chain rule of differentiation: dy / dx = dy / du * du / dx

Constant multiple rule of differentiation: d c f(x) / d x = c d f(x) / d x where c is a real number

dA / dr

= d [1300 (1 + r/12 )⁴⁸] /dr

= 1300 * d [ (1 + r/12 )⁴⁸] /dr (Constant multiple rule)

= 1300 * d [ (1 + r/12 )⁴⁸] /d (1 + r/12 ) * d (1 + r/12 ) / dr (Chain rule, let (1 + r/12 ) be u)

= 1300 * 48 (1 + r/12 )⁴⁷ * (1/12) (Power rule, constant rule)

= 5200 (1 + r/12 )⁴⁷

dA / dr (at r = 4.5%) = 5200 (1 + 4.5%/12 )⁴⁷ = ...

dA / dr (at r = 6.5%) = 5200 (1 + 6.5%/12 )⁴⁷ = ...