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Differentiation is a fundamental concept in calculus. It is used to find the rate at which one quantity changes concerning another. In simple terms, differentiation helps us understand how a function changes as its input changes. For example, when calculating compound interest, we might want to know how the final amount grows as the interest rate changes. This is achieved through differentiation.

The differentiation process involves finding the derivative of a function. The derivative of a function gives us the slope of the function at any point. For compound interest, given by the formula \(A = 1000(1 + i)^3\), we differentiate with respect to the interest rate \(i\). This tells us how the amount \(A\) changes as the interest rate \(i\) changes.

The Chain Rule is a technique used in differentiation when dealing with composite functions. A composite function is made up of two or more functions. For example, in the compound interest formula, \(A = 1000(1 + i)^3\), we have an inner function \(1 + i\) and an outer function \((x)^3\).

When differentiating a composite function, the chain rule states that we first differentiate the outer function and then multiply by the derivative of the inner function. In our exercise, we applied the chain rule as follows:

• Differentiate the outer function \((1 + i)^3\): This gives us \(3(1 + i)^{2}\).
• Differentiate the inner function \(1 + i\): This gives us \1\.

Combining these using the chain rule, we get \(3000(1 + i)^2\).

This result tells us how the compound interest amount \(A\) changes with respect to the interest rate \(i\).

The rate of change is a measure of how one quantity changes in response to another quantity. In the context of our exercise, it refers to how the amount of money accumulated \(A\) changes as the interest rate \(i\) changes. The rate of change is found using differentiation.

For the given compound interest formula, the rate of change is represented by \(\frac{dA}{di} = 3000(1 + i)^2\). This derivative tells us the sensitivity of the amount \(A\) to changes in the interest rate \(i\).

• If the rate of change is high, it means that even a small change in the interest rate \(i\) will result in a significant change in the amount \(A\).
• If the rate of change is low, the amount \(A\) is less sensitive to changes in \(i\).

Understanding the rate of change is crucial for making informed financial decisions, as it helps predict how investments will grow with varying interest rates.