Question

Evaluate the following derivatives. $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right)$$

Short answer

Answer: The derivative of the function \((\ln 2x)^{-5}\) with respect to \(x\) is $$\frac{-5}{x(\ln 2x)^6}$$.

Step-by-step solution

Identify the outer and inner functions.

The outer function is \((u)^{-5}\), where \(u\) is the inner function. The inner function is \(\ln 2x\). Step 2: Find the Derivative of the Outer Function

Find the derivative of the outer function.

The outer function is \((u)^{-5}\), so its derivative with respect to \(u\) is \(-5(u)^{-6}\). Step 3: Find the Derivative of the Inner Function

Find the derivative of the inner function.

The inner function is \(\ln 2x\). The derivative of \(\ln 2x\) with respect to \(x\) is \(\frac{2}{2x} = \frac{1}{x}\). Step 4: Apply the Chain Rule

Apply the chain rule.

Now we apply the chain rule by multiplying the derivative of the outer function with the derivative of the inner function: $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right) = -5(\ln 2x)^{-6} \cdot \frac{1}{x}$$ Step 5: Simplify the Result

Simplify the result.

Combine the terms in the expression to get the final answer: $$\frac{-5}{x(\ln 2x)^6}$$ Thus, the derivative of the given function is $$\frac{-5}{x(\ln 2x)^6}$$

Key concepts

Derivatives

Derivatives are a fundamental concept in calculus. They represent the rate of change of a function as one of its variables changes. In simple terms, a derivative gives us the slope of a function at any given point. When working with derivatives, we often look to find how a function behaves as its input changes, which is crucial in various scientific fields like physics and engineering.

When we say "evaluate the derivative," we're looking to find this instantaneous rate of change. This often involves applying rules and formulas, like the power rule, product rule, or chain rule, to compute the derivative efficiently. In this specific exercise, we're finding the derivative of a function that involves logarithms and powers. This is where these techniques come into play. Mastering derivatives not only helps in solving mathematical problems but also in understanding the real-world phenomena that these functions model.

Logarithmic Functions

Logarithmic functions are inverse functions of exponential functions. They help in simplifying complex multiplication and division problems into manageable addition and subtraction. The natural logarithm \((\ln)\) is especially important, as it uses the Euler's number \((e)\), a key constant in mathematics. In this exercise, the logarithmic function \(\ln 2x\) is an important part of the problem. This is our inner function, which we must differentiate to apply the chain rule.

To differentiate \(\ln 2x\), we use the rule that the derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\). So, for \(\ln 2x\), we find the derivative as \(\frac{1}{x}\). Logarithmic differentiation is extremely useful, especially when dealing with complex products or quotients in functions. Understanding this concept is pivotal when faced with composite functions that contain logarithms.

Outer and Inner Functions

In mathematics, particularly in calculus, we often deal with composite functions. These are functions that are made by combining two or more functions. A common scenario is when you have an inner function nested within an outer function, like in this exercise. We have \(\ln 2x\) as the inner function and \((u)^{-5}\) as the outer function where \(u = \ln 2x\).

Understanding outer and inner functions is crucial when using the chain rule, a powerful tool in calculus. The chain rule helps us find the derivative of composite functions. Essentially, it tells us to multiply the derivative of the outer function by the derivative of the inner function. In this exercise, this process is what allows us to find the correct derivative efficiently. Mastering the identification and differentiation of these nested functions makes tackling more complex derivative problems manageable.