In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))

Verified step by step guidance

1. Identify the function given: \(y = \ln(\ln(x))\). This is a composition of two functions: the natural logarithm of another natural logarithm.
2. Apply the chain rule for derivatives. The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
3. Set the outer function as \(f(u) = \ln(u)\) and the inner function as \(u = \ln(x)\). Then, find the derivative of the outer function with respect to \(u\): \(f'(u) = \frac{1}{u}\).
4. Find the derivative of the inner function \(u = \ln(x)\) with respect to \(x\): \(u' = \frac{1}{x}\).
5. Combine these results using the chain rule: \(\frac{dy}{dx} = \frac{1}{\ln(x)} \cdot \frac{1}{x}\). This expresses the derivative of \(y\) with respect to \(x\).

Key Concepts

Chain Rule

The chain rule is a fundamental technique for finding the derivative of composite functions. It states that the derivative of a function composed of two functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Derivative of the Natural Logarithm Function

The derivative of the natural logarithm function ln(x) with respect to x is 1/x, provided x > 0. This rule is essential when differentiating expressions involving logarithms, especially nested logarithms like ln(ln(x)).

Domain Considerations for Logarithmic Functions

When differentiating logarithmic functions, it's important to consider the domain where the function is defined. For y = ln(ln(x)), both ln(x) and x must be positive, so x > 1. This ensures the function and its derivative are valid and real-valued.