Problem 55 - Calculus of a Single Variable

Question

In Exercises 41–64, find the derivative of the function. $$ y=\ln \left(\ln x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative of the given function is \(y' = \frac{2}{x\ln(x^{2})}\)

Step 1: Differentiate the outer function

intake the derivative of the outermost function which is \(y=\ln(u)\) here. The derivative of the natural log of u (where u is a function of x) is \(1/u\). Here \(u=\ln(x^{2})\), so the derivative becomes \(1/\ln(x^{2})\). But don't forget to then differentiate \(\ln(x^{2})\) due to chain rule.

Step 2: Apply the Chain Rule

Now you need to find the derivative of \(u=\ln(x^{2})\), see it as \(v=\ln(w)\) where \(w=x^{2}\). Its derivative is \(1/w = 1/x^{2}\). But again don't forget to differentiate \(x^{2}\) due to the chain rule.

Step 3: Calculating the final derivative of the function

Finally, calculate the derivative of \(x^{2}\) which is simply \(2x\). After calculating these, you can now write down the whole derivative as it is product of the three derivatives due to chain rule, which gives us the derivative of \(y= \ln(\ln(x^{2}))\) to be \(y' = \frac{1}{\ln(x^{2})} \times \frac{1}{x^{2}} \times 2x = \frac{2}{x\ln(x^{2})}\)

Key Concepts

Natural Logarithm Derivative

Understanding the derivative of the natural logarithm is essential in calculus. The natural logarithm function, denoted by \( \text{ln}(x) \), represents the logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The derivative of \( \text{ln}(x) \) with respect to \( x \) is unique because it simplifies to \( 1/x \).

This simplification is based on the definition of \( e \) and the properties of logarithms. To derive a natural logarithmic function that is more complex, such as \( \text{ln}(g(x)) \) where \( g(x) \) is another function of \( x \), we must use the chain rule. The chain rule allows us to differentiate composite functions by taking the derivative of the outer function (in this case, \( \text{ln}(x) \)) and multiplying it by the derivative of the inner function \( g(x) \).

For the exercise problem \( y = \text{ln}(\text{ln}(x^2)) \), we start by differentiating the outer \( \text{ln} \) function, obtaining \( 1/u \) where \( u = \text{ln}(x^2) \). Understanding this foundational concept helps in grasping more complex derivatives involving the natural logarithm.

Implicit Differentiation

Implicit differentiation is a technique used when we work with equations where the dependent variable cannot be easily solved for and expressed explicitly as a function of the independent variable. This situation often occurs in equations defining curves or shapes that are not functions in the traditional sense.

In this approach, we differentiate both sides of an equation with respect to the independent variable, and through this process, the derivative of the dependent variable emerges. Throughout this differentiation step, we apply derivative rules such as product rule, quotient rule, and the chain rule as appropriate. Implicit differentiation is particularly useful when dealing with functions involving \( \text{ln}(x) \) and more complex expressions that involve natural logs of functions, as seen in our exercise example.

By using implicit differentiation, we can handle the derivative of the composite function \( \text{ln}(x^2) \) within the larger function \( y = \text{ln}(\text{ln}(x^2)) \), leading to a clear and methodical solution. This differentiation method enhances our toolkit for tackling diverse calculus problems.

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives beyond the first derivative of a function. The second derivative, or the derivative of the first derivative, can convey information about the concavity of the original function and the acceleration if the function describes motion. Third-order derivatives and beyond can provide insights into changes in acceleration and other more complex behaviors.

Calculating higher-order derivatives typically involves applying the basic differentiation rules repeatedly. When a function involves composite functions, such as in our exercise with \( y = \text{ln}(\text{ln}(x^2)) \), each differentiation stage might require the use of the chain rule multiple times. To find second, third, or even fourth derivatives of such functions, we continue the pattern established by the chain rule and pay close attention to how each function component impacts the derivative.

The concept of higher-order derivatives is important because it gives a complete picture of the behavior of functions and can be critical for understanding the complexity of curves in graphing, the motion of objects in physics, or the optimization of systems in engineering.