Question

Derivatives of logarithmic functions Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing \(f^{\prime}(x)\). $$y=\log _{2}\left(\log _{2} x\right)$$

Short answer

Question: Find the derivative of the function \(y = \log_2(\log_2 x)\). Answer: The derivative of the given function is \(y^{\prime}(x) = \frac{1}{x\log_2 x(\ln{2})^2}\).

Step-by-step solution

Identify the functions

The function provided is a composition of two logarithmic functions. First, we need to identify these functions: 1. \(u(x) = \log_2 x\) 2. \(y(x) = \log_2(u)\) Now we have \(y(x) = \log_2(\log_2 x) = \log_2(u(x))\).

Find the derivative of each function

Now we need to find the derivatives for both \(u(x)\) and \(y(x)\). 1. For \(u(x) = \log_2 x\), we can find the derivative \(u^\prime(x)\) as: $$u^{\prime}(x) = \frac{1}{x\ln{2}}$$ 2. For \(y(x) = \log_2{u}\), we can similarly find the derivative \(y^\prime(u)\) as: $$y^{\prime}(u) = \frac{1}{u\ln{2}}$$

Use the chain rule

We will now use the chain rule to find the overall derivative \(y^\prime(x)\). The chain rule states that if \(y\) is a function of \(u\), and \(u\) is a function of \(x\) (as in this case), then: $$y^{\prime}(x) = y^{\prime}(u) \cdot u^{\prime}(x)$$ Plugging in the values we calculated for \(y^{\prime}(u)\) and \(u^{\prime}(x)\), we have: $$y^{\prime}(x) = \frac{1}{u\ln{2}} \cdot \frac{1}{x\ln{2}} = \frac{1}{xu(\ln{2})^2} $$

Replace u(x) with the original function

Now we need to substitute \(u(x)\) back into our expression for \(y^{\prime}(x)\): $$y^{\prime}(x) = \frac{1}{xu(\ln{2})^2} = \frac{1}{x\log_2 x(\ln{2})^2}$$

Final Answer

The derivative of the given function \(y(x) = \log_2(\log_2 x)\) is: $$ y^{\prime}(x) = \frac{1}{x\log_2 x(\ln{2})^2} $$

Key concepts

Chain Rule

The chain rule is an essential concept in calculus, particularly when dealing with derivatives of composite functions. In essence, it allows us to unpack the derivative of a function that is nested within another. Think of it like peeling an onion one layer at a time. If we have a function \( y = f(g(x)) \), the chain rule states that the derivative is \( y' = f'(g(x)) \cdot g'(x) \).

Let's imagine you're taking a snapshot: \( f \) is the camera, \( g \) is the scenery, and \( x \) is the light. To capture the effect of changing light on the picture, you'd first see how the scenery changes with light and then how the camera captures that change. Similarly, in calculus, the chain rule helps us understand how a change in \( x \) affects \( y \) by considering the intermediate function \( g \). This process is critical when dealing with derivatives of logarithmic functions that involve multiple layers of operations.

Logarithmic Differentiation

Logarithmic differentiation is a technique that can make finding derivatives of complex functions more manageable. This approach involves taking the logarithm of a function and then using the differentiation rules for logarithms. It is exceptionally useful when functions involve products, quotients, or powers that are themselves functions.

By transforming the original function into a logarithmic form, we are able to utilize the properties of logarithms to simplify the differentiation process. Once we've taken the logarithm of the function, we differentiate both sides with respect to \( x \) and then solve for the derivative of the original function. It's like decoding a complex cipher; logarithmic differentiation translates a hard-to-solve problem into a language of calculus that we can easily work with.

Properties of Logarithms

To truly grasp logarithmic differentiation, one must understand the properties of logarithms. These properties are rules that allow for the simplification and manipulation of logarithmic expressions. Some of the key properties include the product rule, quotient rule, and power rule, which state respectively that:

• \(\log_b(MN) = \log_b(M) + \log_b(N)\)
• \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
• \(\log_b(M^p) = p\cdot\log_b(M)\)

These properties can transform a logarithm of a product into a sum of logarithms, a quotient into a difference, and a power into a multiplication. They're like a Swiss Army knife for logarithmic equations, enabling us to refactor intricate expressions into simpler terms, making differentiation more straightforward.

Composition of Functions

When we calculate the derivative of a function that is itself defined by another function, we're dealing with a composition of functions. It means we have a function inside another function—like a Russian nesting doll. In our exercise, \( \log_2(\log_2 x) \) is such a composition with \( u(x) = \log_2 x \) nested inside another logarithmic function.

Understanding the composition of functions is fundamental for applications of the chain rule. It's the groundwork that allows us to differentiate more complex equations by breaking them down into simpler parts that we can handle individually before putting them back together. Mastering this concept is like building with LEGO: you work with individual pieces to create an intricate whole.