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The chain rule is a fundamental concept in calculus, especially when you deal with composite functions. A composite function is essentially a function within another function. For instance, in our exercise, the function \(f(t)=\sqrt{3t^2-t}\) is composed of two functions: the outer square root function and the inner polynomial function \(3t^2-t\).
The chain rule helps us find the derivative of such composite functions efficiently. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function itself. In simple terms:

• Differentiate the outer function.
• Evaluate it with the inner function unchanged.
• Multiply this result by the derivative of the inner function.

Using this rule is crucial as it compartmentalizes derivative calculation, making it methodical and error-free. Remember, the chain rule formula is neatly expressed as:\[\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)\] This methodical approach simplifies even the most tangled functions.

The power rule is a straightforward and commonly used rule in calculus to find the derivative of functions expressed as powers of \(x\). It states that if you have a function \(f(x) = x^n\), the derivative will be \(f'(x) = nx^{n-1}\). In the exercise, we initially rewrote the square root function as a power: \(f(t) = (3t^2 - t)^{1/2}\). By doing so, applying the power rule becomes more apparent, and it aids in the stepwise differentiation process.

• Determine the power of the expression.
• Multiply by the power.
• Decrease the power by one.

This process gives us the derivative of the outer function with respect to its argument, which in this case was \((3t^2 - t)\). The power rule is essential for breaking down complexities into manageable parts, simplifying tasks like employing the chain rule further down the line.

Identifying outer and inner functions is a critical step when using the chain rule. It involves a keen look at the structure of your function to separate it into layers.
In the provided exercise, we define:

• The outer function as the square root, \(f(u) = u^{1/2}\), after rewriting the original square root as a power for easier differentiation.
• The inner function is the polynomial \(g(t) = 3t^2 - t\) that resides within the outer square root function.

Understanding which part is your inner function and which is your outer function smooths the way for applying the chain rule. It ensures each part is correctly differentiated. This layered understanding decouples complex function differentiation into simpler tasks, thereby reducing potential mistakes during calculations.