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The chain rule is a powerful tool in calculus for finding the derivative of composite functions. When you have a function inside another function, like in the exercise with \( g(t) = \ln(\ln t) \), understanding how to use the chain rule is essential.

Here's the basic idea of the chain rule: if you have two functions, say \( f \, \text{and} \, g \), where \( f(g(x)) \) is a composition of these two functions, its derivative is given by the formula:

• Find the derivative of the outer function \( f \), evaluating it at the inner function \( g(x) \).
• Find the derivative of the inner function \( g \).
• Multiply these two derivatives.

So, for \( g(t) = \ln(\ln t) \), imagine \( g \) as the outer function and \( \ln t \) as the inner function. Differentiate each, and then multiply them as per the chain rule. This method helps in tackling complex derivatives efficiently.

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828. It is a common mathematical function used in calculus, especially when dealing with exponential growth.

• The natural logarithm is useful because of its unique properties, such as \( \ln(e^x) = x \).
• In calculus, one fundamental property is that the derivative of \( \ln x \) is \( \frac{1}{x} \).
• This makes the natural logarithm a powerful tool in differentiation, simplifying calculations significantly.

In the original exercise, \( \ln \) appears twice: first as \( \ln t \) and then as \( \ln \) again, creating a situation where understanding the properties and derivatives of the natural logarithm becomes crucial.

Composition of functions involves inserting one function inside another, creating a new function. When differentiating such compositions, as seen in the exercise with \( g(t) = \ln(\ln t) \), this can initially seem daunting.

To get a clear picture, consider this: if \( g(t) = f(u) \) and \( u = h(t) \), then \( g(t) = f(h(t)) \) is a composition. Here, it's essential to:

• Identify the outer function (\( f \)) and the inner function (\( h \)).
• Evaluate the composite function step by step.
• Apply differentiation rules through careful observation.

In the case of \( g(t) = \ln(\ln t) \), \( \ln t \) is first computed, and then the natural logarithm of that result is taken. Recognizing this layering makes applying the chain rule straightforward, as you carefully handle each function in its order within the composition.