The value of invesment is .

(a).

Principal Income is  dollars.

Rate of interest  .

Time period is  years.

(i).

Find the compound interest compounded annually.

Compound interest is .

For annually compounded .

Substtitute all the above values into .

Compound interest compounded annually : .

(ii).

Find the compound interest compounded semi annualy.

For semi annualy compounded .

Substtitute all the above values into .

Compound interest compounded semi annualy : .

(iii).

Find the compound interest compounded monthly.

For monthly compounded .

Substtitute all the above values into .

Compound interest compounded monthly : .

(iv)

Find the compound interest compounded weekly.

For weekly compounded .

Compound interest compounded weekly : .

(v).

Find the compound interest compounded weekly.

For daily compounded .

Compound interest compounded daily:.

(vi).

Find the compound interest compounded continuously.

Continuously compound interest .

Compound interest compounded continuously : .

(b).

Compounding interest .

Substitute  and .

Initially .

.

The equation is .

Apply derivative on each side with respect to .

Substitute  and  in the above equation.

.

(a).

(i) Compound interest compounded annually : .

(ii) Compound interest compounded semi annualy : .

(iii) Compound interest compounded monthly : .

(iv) Compound interest compounded weekly : .

(v) Compound interest compounded daily:.

(vi) Compound interest compounded continuously : .

(b).

 and  .