Question

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

Short answer

The derivative is \( \frac{k}{x} \) using both methods.

Step-by-step solution

Identify the Function

We have the function \( y = \ln \left( x^k \right) \). We need to find its derivative.

Rewrite Using the Logarithm Rule

Using the logarithm rule \( \ln(a^p) = p \ln(a) \), we can rewrite the function as \( y = k \ln(x) \).

Differentiate Using Basic Derivative Rules

The derivative of \( y = k \ln(x) \) with respect to \( x \) is \( \frac{dy}{dx} = k \cdot \frac{1}{x} = \frac{k}{x} \).

Apply the Chain Rule to the Original Function

Consider the original function \( y = \ln(u) \) where \( u = x^k \). By chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Differentiate \( \ln(u) \) with Respect to \( u \)

The derivative of \( \ln(u) \) is \( \frac{1}{u} \).

Differentiate \( u = x^k \) with Respect to \( x \)

The derivative \( \frac{du}{dx} \) for \( u = x^k \) is \( kx^{k-1} \).

Compute the Entire Derivative Using Chain Rule

Plug the results into the chain rule formula: \( \frac{dy}{dx} = \frac{1}{u} \cdot kx^{k-1} = \frac{kx^{k-1}}{x^k} \).

Simplify the Solution from the Chain Rule

Simplify \( \frac{kx^{k-1}}{x^k} = \frac{k}{x} \).

Verify Both Methods Yield Equivalent Results

Both methods produce \( \frac{dy}{dx} = \frac{k}{x} \), indicating the solution is correct.

Key concepts

Chain Rule

In calculus, the chain rule is an essential formula for computing the derivative of a composite function. Imagine you have a function that is made by plugging one function into another. This rule helps you find the derivative by using the derivatives of the inside and outside functions.

To apply the chain rule, follow these steps:

• Identify the outer and inner functions. For example, if you have \( y = f(g(x)) \), then \( f \) is the outer function and \( g \) is the inner.
• Take the derivative of the outer function, treating the inner function as a constant.
• Multiply by the derivative of the inner function.

For the given exercise, consider \( y = \ln(u) \) where \( u = x^k \). The derivative of the outer function \( \ln(u) \) is \( \frac{1}{u} \). Next, differentiate the inner function \( u = x^k \), which gives \( kx^{k-1} \). Multiply them to get \( \frac{kx^{k-1}}{x^k} \), which simplifies neatly to \( \frac{k}{x} \).

This step-wise approach through the chain rule reinforces understanding on computing derivatives of complex functions.

Logarithm Rule

The logarithm rule is a handy shortcut when you need to rewrite logarithmic expressions, especially those involving powers. This rule states \( \ln(a^p) = p \ln(a) \). This makes differentiation straightforward when dealing with exponents.

In this exercise, the function \( \ln(x^k) \) can be rewritten using this rule. Just take the exponent \( k \) in front of the logarithm, making it \( k \ln(x) \). This transformation simplifies the differentiation process, as you're now dealing with a simpler expression.

Once simplified to \( y = k \ln(x) \), differentiating is basic. When you differentiate \( k \ln(x) \), treat \( k \) as a constant factor, giving \( \frac{k}{x} \).

The logarithm rule not only simplifies complex expressions but also makes differentiation clearer and more manageable.

Differentiation

Differentiation is the central concept of calculus that involves finding the rate at which a function changes at any given point. It's like asking: "How steep is the curve at this specific moment?"

To differentiate, you apply different rules based on the function you're dealing with. Some basic rules include the power rule, product rule, quotient rule, and of course, the chain rule—the latter being crucial for this exercise.

In differentiating \( \ln(x^k) \), once it is rewritten as \( k \ln(x) \), you are in familiar territory. Using the basic rule \( \frac{d}{dx} \ln(x) = \frac{1}{x} \), and recognizing \( k \) as a constant, results in \( \frac{k}{x} \).

Differentiation allows you to find derivatives efficiently and draw meaningful insights about the behavior of functions.

Function Transformation

Function transformation involves changing the form of a function to make it easier to analyze or differentiate. Here, the transformation primarily applies to logarithmic functions using properties like the logarithm rule.

By transforming \( \ln(x^k) \) into \( k \ln(x) \) using the logarithm rule, the task of differentiation is greatly simplified. When functions take on transformed forms, they often reveal patterns or behaviors that weren't initially obvious.

This transformation technique can help in multiple calculus scenarios:

• Rewriting functions into a standard form for easier application of derivative rules.
• Extracting constants out of logarithmic expressions, facilitating simpler differentiation.
• Uncovering simpler derivatives through substitution and transformation methods.

Function transformation is a powerful tool in calculus, driving home the important lesson that when functions look complex, transformation might be the key.