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In calculus, the concept of the first derivative is fundamental for understanding any function's rate of change. When we take the first derivative of a function, we are essentially finding how quickly the function's output is changing concerning its input. For our example, the function \(n(x) = -1 + 8x + 2x^2 - 0.4x^3\) represents the number of new products based on the R&D budget \(x\). Let's break down the process of finding the first derivative:

• The first derivative of each term is derived using the power rule. The power rule states that if you have a term in the form of \(ax^n\), the derivative will be \(n \cdot ax^{n-1}\).
• Applying this, we get \(n'(x) = 8 + 4x - 1.2x^2\).

This derivative, \(n'(x)\), tells us how the number of new products changes as the R&D budget changes.
For example, a positive derivative at a certain value of \(x\) indicates that investing more in R&D tends to increase the number of new products.

The second derivative provides information on the curvature of a function. It helps us understand whether the function is curving upwards or downwards, in other words, if it is concave up or concave down. This is incredibly useful when analyzing rates of change more deeply.For the example function \(n(x) = -1 + 8x + 2x^2 - 0.4x^3\):

• The second derivative, found by differentiating \(n'(x) = 8 + 4x - 1.2x^2\), is \(n''(x) = 4 - 2.4x\).

Evaluating this at specific points:

• At \(x = 1\), \(n''(1) = 1.6\), which is positive, indicating the function is concave up. This suggests an increasing rate of new product development.
• At \(x = 3\), \(n''(3) = -3.2\), which is negative, indicating the function is concave down. This means the rate of development decreases as the budget increases beyond a certain point.

This analysis is crucial for businesses to understand when and how they benefit from increased spending.
The behavior of the second derivative guides companies in making efficient budget decisions.

Diminishing returns is a key concept in economics and business, often examined using calculus. It refers to a point at which the rate of profit or benefit gained is less than the amount of money or energy invested.In the context of our function, the diminishing returns occur when the second derivative changes from positive to negative. This point indicates where increasing the budget starts to yield less additional benefit in terms of new product development.For our function, \(n''(x) = 4 - 2.4x\):

• Setting \(n''(x) = 0\) gives the equation \(4 - 2.4x = 0\),
• Solving for \(x\) yields \(x = \frac{5}{3}\).

This implies that an R&D budget of approximately \(\frac{5}{3}\) thousand dollars represents the threshold where the company begins to experience diminishing returns.
Past this budget point, each additional dollar spent results in a lesser rate of new product creation. This insight helps optimize budget allocations, ensuring that investments are made where they are most effective.