Suppose that the average yearly cost per item for producing...

Suppose that the average yearly cost per item for producing $x$ items of a business product is $\bar{C}(x)=10+\frac{100}{x} .$ If the current production is $x=10$ and production is increasing at a rate of 2 items per year, find the rate of change of the average cost.

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Key Concepts

Related Rates

This concept involves finding the rate at which one quantity changes with respect to time by relating it to another quantity whose rate of change is known. It is particularly useful when two or more variables that depend on time are connected by an equation, and you need to determine how a change in one affects the others.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When a variable depends on another variable that in turn depends on time, the chain rule allows you to link the derivative with respect to time by multiplying the derivative of the outer function with the derivative of the inner function.

Instantaneous Rate of Change

This represents the derivative of a function at a specific point, providing a measure of how the function is changing at that exact moment. It is essential for understanding the behavior of dynamic systems and is often used in real-world problems to analyze and predict immediate impacts of changing quantities.

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