91Ó°ÊÓ

Calculus is a branch of mathematics that focuses on change. It helps us understand phenomena involving rates of change and accumulation. The field is divided into two main areas:

• Differential calculus, which deals with the concept of the derivative and the rate of change of functions.
• Integral calculus, which focuses on the accumulation of quantities and areas under curves.

In our problem, we are dealing with differential calculus, essentially finding how a function changes as its input changes. This involves finding the derivative of the given function.

When differentiating functions in calculus, especially composite functions, the chain rule is a powerful tool. The chain rule helps us find the derivative of a function composed of other functions. The rule states:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \times g'(x) \]
This means you first differentiate the outer function while keeping the inner function the same. Then multiply by the derivative of the inner function.

In our example, to differentiate the functions \(u(x) = (3.21x - 5.87)^3\) and \(v(x) = (2.36x - 5.45)^5\), we applied the chain rule as follows:

• For \(u(x)\), we differentiated the outer function \((3.21x - 5.87)^3\) and then multiplied it by the derivative of the inner function \(3.21x - 5.87\).
• We repeated this process with \(v(x)\), differentiating \((2.36x - 5.45)^5\) and multiplying it by the derivative of \(2.36x - 5.45\).

A derivative represents how a function changes as its input changes. It's the central concept in differential calculus. If a function \(y = f(x)\) changes at a rate, the derivative \(f'(x)\) gives us that rate of change.

To find the derivative of a product of functions, we use the product rule. The product rule states that the derivative of a product of two functions \(u(x)\) and \(v(x)\) is:
\[ \frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x) \]
In our problem, we have

• \(u(x) = (3.21 x - 5.87)^3\)
• \(v(x) = (2.36 x - 5.45)^5\)

After finding their individual derivatives using the chain rule, we combine them as per the product rule to get the final derivative:
\[ r'(x) = 9.63 (3.21 x - 5.87)^2 (2.36 x - 5.45)^5 + (3.21 x - 5.87)^3 \times 11.8 (2.36 x - 5.45)^4 \] This resulting expression gives us the rate of change of the original function \(r(x)\).