Question

Compute \(\frac{d}{d x}(\ln (k x))\) two ways: (a) Using the Chain Rule, and (b) by first using the logarithm rule \(\ln (a b)=\ln a+\ln b\), then taking the derivative.

Short answer

The derivative is \(\frac{1}{x}\) for both methods.

Step-by-step solution

Differentiate Using the Chain Rule

To differentiate \(\ln(kx)\) using the Chain Rule, first recognize that the composite function is of the form \(\ln(u)\) where \(u = kx\). The derivative of \(\ln(u)\) is \(\frac{1}{u}\cdot \frac{du}{dx}\). Here, \(\frac{du}{dx} = k\), so the derivative is \(\frac{1}{kx} \cdot k\), which simplifies to \(\frac{1}{x}\).

Apply the Logarithm Rule

Use the logarithm identity \(\ln(ab) = \ln a + \ln b\) to rewrite \(\ln(kx)\) as \(\ln k + \ln x\).

Differentiate Each Term Separately

Differentiate the expression \(\ln k + \ln x\). The derivative of \(\ln k\) is 0 because \(k\) is a constant. The derivative of \(\ln x\) is \(\frac{1}{x}\). Therefore, the overall derivative is \(0 + \frac{1}{x} = \frac{1}{x}\).

Compare Both Results

Both methods result in the same derivative, \(\frac{1}{x}\), confirming the computations are correct.

Key concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus, particularly when dealing with composite functions. It helps us find the derivative of a function that is composed of another function. In simpler terms, if you have a function nested within another function, the Chain Rule is your go-to method for differentiation.

For example, consider the problem of differentiating \(\ln(kx)\), where \(kx\) is a function within a function. Here, the outer function is \(\ln(u)\), and the inner function is \(u = kx\). The Chain Rule states that to differentiate \(\ln(u)\), we first find the derivative of the outer function \(\frac{1}{u}\), and then multiply it by the derivative of the inner function \(u\).

In this case, since the derivative of \(kx\) with respect to \(x\) is \(k\), the final derivative becomes \(\frac{1}{kx} \cdot k\), simplifying to \(\frac{1}{x}\). The Chain Rule breaks the differentiation down into manageable steps by isolating each part of the function.

Logarithm Rule

The Logarithm Rule is an essential technique in simplifying logarithmic expressions before differentiation. This rule is particularly handy in breaking down complex logarithmic functions into simpler terms by using logarithmic identities.

Take the identity \(\ln(ab) = \ln a + \ln b\), which allows us to rewrite a product inside a logarithm as a sum of logarithms. When you apply this rule to the function \(\ln(kx)\), it becomes \(\ln k + \ln x\). This transformation is crucial because it separates the constant term from the variable term, making differentiation straightforward.

Now, differentiating \(\ln k + \ln x\):

• \(\ln k\) is a constant, so its derivative is 0.
• \(\ln x\) is the variable part, with a derivative of \(\frac{1}{x}\).

The overall derivative results in \(0 + \frac{1}{x} = \frac{1}{x}\). This method shows how the Logarithm Rule can simplify the differentiation process.

Composite Function

Composite functions occur when one function is applied to the result of another function. They are commonly represented in the form \(f(g(x))\), where \(g(x)\) is the inner function and \(f\) is the outer function. Understanding how to handle composite functions is vital in calculus to correctly differentiate them.

In the context of \(\ln(kx)\), the function is composite because \(ln\) operates on \(kx\). For such functions, knowing the derivative of the outer and inner functions is critical. This understanding simplifies the application of rules like the Chain Rule.

Learning to separate a composite function into its components allows us to manage and solve complex calculus problems. Recognizing composite functions helps in selecting the appropriate technique and simplifies the computational process.

Derivative of Logarithm

The derivative of the natural logarithm \(\ln x\) is a fundamental concept in calculus. It is important to know that \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), a rule that comes in handy in solving many problems involving logarithmic functions.

When differentiating logarithmic expressions, having the derivative of \(\ln x\) as a reference can speed up calculations and make the process more intuitive. For instance, when you break down \(\ln(kx)\) using the Logarithm Rule, you immediately apply \(\frac{1}{x}\) to the term \(\ln x\).

This consistency underscores the importance of mastering basic derivatives like \(\ln x\). By solidifying your grasp of the derivative of logarithms, you pave the way for handling more sophisticated calculus challenges confidently.