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The chain rule is a fundamental concept in differential calculus. It allows us to differentiate composite functions. Simply put, a composite function is a function inside another function. To apply the chain rule, differentiate the outer function and multiply it by the derivative of the inner function.

Consider the function given in the exercise: \( \mathbf{r}_{1}(2t) \). The outer function is \( \mathbf{r}_{1}(x) \) and the inner function is \( 2t \). Applying the chain rule, we first differentiate the outer function with respect to the inner function to get \( \mathbf{r}_{1}'(2t) \). Then, we multiply by the derivative of the inner function, which is 2. Therefore, the derivative becomes \( 2 \mathbf{r}_{1}'(2t) \).

Applying the chain rule helps in solving complex differentiation problems by breaking them down into simpler parts. It is especially useful when dealing with functions that are not straightforward to differentiate directly.

Derivatives are a key concept in calculus. They measure how a function changes as its input changes. In simpler terms, a derivative gives the rate at which one quantity changes with respect to another. This is essential for understanding motion, rates of change, and even optimization problems.

In the exercise, we calculated two derivatives using the chain rule. Taking the derivative of \( \mathbf{r}_{1}(2t) \), where 2t is the argument, and \( \mathbf{r}_{2}\left(\frac{1}{t}\right) \), where \( \frac{1}{t} \) is the argument. For the latter, we found that the derivative of \( \frac{1}{t} \) is \( -\frac{1}{t^2} \). Therefore, the term \( \mathbf{r}_{2}\left(\frac{1}{t}\right) \) was successfully differentiated as \( -\frac{1}{t^2} \mathbf{r}_{2}'\left(\frac{1}{t}\right) \).

Derivatives allow us to predict how a function behaves, analyze trends, and solve real-world problems efficiently. Mastering the basics of derivatives is crucial for advanced calculus problem-solving.

Solving calculus problems involves understanding and applying various fundamental rules and concepts. A systematic approach can make difficult problems easier to tackle. Here is how you can approach solving problems like the one in the exercise:

• Break down the problem by identifying the different parts of the function that need differentiation.
• Determine which rules apply to each part, such as the chain rule, product rule, or quotient rule.
• Differentiate each part separately and simplify the result.
• Combine the differentiated parts to arrive at the final solution.

These steps were followed exactly in the given exercise solution. You should always take time to verify your results, just as in the final step of the exercise where the derivative was checked against the original problem statement. This ensures accuracy and reinforces the learning process.