Problem 67 - Calculus for Biology and Medicine

Question

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{\ln x} $$

Step-by-Step Solution

Answer

The first derivative is \( f'(x) = x^{\ln x - 1} \cdot 2 \ln x \).

Step 1: Taking the Natural Logarithm

Take the natural logarithm on both sides of the given function.\[ \ln f(x) = \ln(x^{\ln x}) \]Using the properties of logarithms, simplify the expression:\[ \ln f(x) = \ln x \cdot \ln x = (\ln x)^2 \]

Step 2: Differentiate Implicitly

Differentiate both sides with respect to \(x\).On the left side, we have:\[ \frac{1}{f(x)} \cdot f'(x) \]On the right side, use the chain rule for differentiation:\[ \frac{d}{dx}((\ln x)^2) = 2 \cdot \ln x \cdot \frac{1}{x} \]Putting it together:\[ \frac{1}{f(x)} \cdot f'(x) = \frac{2 \ln x}{x} \]

Step 3: Solve for \(f'(x)\)

Multiply both sides by \(f(x)\) to isolate \(f'(x)\):\[ f'(x) = f(x) \cdot \frac{2 \ln x}{x} \]Since we know that \(f(x) = x^{\ln x}\), substitute back:\[ f'(x) = x^{\ln x} \cdot \frac{2 \ln x}{x} \]Simplifying gives:\[ f'(x) = x^{\ln x - 1} \cdot 2 \ln x \]

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used in calculus to find the derivative of functions that are not explicitly solved for one variable in terms of another. This can be especially useful when dealing with complex equations.
When applying implicit differentiation, you differentiate each term with respect to the independent variable. In the problem above, since the left side of the equation involves a function of the form \( \ln f(x) \), we take the derivative of this expression implicitly by considering \( f(x) \) as a function of \( x \).

• Differentiate the logarithmic expression on the left-hand side to get \( \frac{1}{f(x)} \cdot f'(x) \).
• This step involves recognizing that \( f'(x) \) is the derivative of \( f(x) \) with respect to \( x \), which we are trying to solve for.

Thus, implicit differentiation allows us to find the derivative of more complex functions, making it a powerful tool in calculus.

Chain Rule

The chain rule is a fundamental concept in differentiation used when dealing with composite functions. It states that if you have a function that is a composition of two functions, such as \( h(x) = g(f(x)) \), the derivative is: \( h'(x) = g'(f(x))\cdot f'(x) \).
In the solution provided, the chain rule is applied to differentiate \( (\ln x)^2 \). Since \( (\ln x)^2 \) is a composition of the square function and the natural logarithm function, we apply the chain rule:

• First, differentiate the outer function, which is \( u^2 \) with respect to \( u \), resulting in \( 2u \).
• Here, \( u \) is \( \ln x \), so next, differentiate \( \ln x \) with respect to \( x \), resulting in \( \frac{1}{x} \).
• Combine these results to get \( 2 \cdot \ln x \cdot \frac{1}{x} \) for the derivative of \( (\ln x)^2 \).

The chain rule effectively helps in breaking down complex derivatives into simpler parts, enabling easier differentiation of composite functions.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. It's a crucial concept in calculus, especially when working with continuous growth models and solving logarithmic differentiation problems.

• The natural logarithm is particularly useful for simplifying exponential expressions in differentiation, as seen in the given problem. By taking the natural logarithm of both sides, we convert the original expression \( x^{\ln x} \) into a more manageable form \( (\ln x)^2 \), which can be differentiated more straightforwardly.
• When handling functions involving logarithms, remember that \( \ln x \) differentiates to \( \frac{1}{x} \).
• Using properties of logarithms, such as \( \ln(a^b) = b\cdot\ln a \), allows for the simplification of expressions to facilitate easier differentiation.

The natural logarithm is thus a powerful tool in calculus, often simplifying complex problems into equations that are easier to work with.