Problem 62 - Calculus for Biology and Medicine

Question

Assume that \(f(x)\) is differentiable with respect to \(x\). Show that $$\frac{d}{d x} \ln \left[\frac{f(x)}{x}\right]=\frac{f^{\prime}(x)}{f(x)}-\frac{1}{x}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{f'(x)}{f(x)} - \frac{1}{x} \).

Step 1: Understand the Objective

We need to find the derivative of the function \( \ln \left[ \frac{f(x)}{x} \right] \) with respect to \( x \). Differentiation is the process of finding the rate of change of a function.

Step 2: Apply the Logarithm Property

Use the properties of logarithms to simplify the expression before taking the derivative. Recall that \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \). Thus, \( \ln \left[ \frac{f(x)}{x} \right] = \ln(f(x)) - \ln(x) \).

Step 3: Differentiate Logarithmic Expressions

Differentiate both parts of the expression \( \ln(f(x)) - \ln(x) \) with respect to \( x \). Use the rule that the derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{u'}{u} \), where \( u \) is a function of \( x \).

Step 4: Differentiate \( \ln(f(x)) \)

The derivative of \( \ln(f(x)) \) with respect to \( x \) is \( \frac{f'(x)}{f(x)} \). This follows from the chain rule and the property of logarithms used for differentiation.

Step 5: Differentiate \( \ln(x) \)

The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). This is a standard result for differentiating the natural log of \( x \).

Step 6: Combine the Derivatives

Combine the results from Steps 4 and 5. The derivative of the entire expression \( \ln(f(x)) - \ln(x) \) is \( \frac{f'(x)}{f(x)} - \frac{1}{x} \).

Step 7: Conclusion

Through the differentiation and application of logarithm properties, we have shown that \( \frac{d}{dx} \ln \left[ \frac{f(x)}{x} \right] = \frac{f'(x)}{f(x)} - \frac{1}{x} \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. Think of it as the mathematical way to inform us how steep a hill is if our function were representing a mountain range. The key tool used for differentiation is the derivative. When you take the derivative of a function, you are essentially calculating how the function's output value changes as its input value changes.

For example, if you have a function like \( f(x) = x^2 \), the derivative, denoted as \( f'(x) \) or \( \frac{df}{dx} \), tells us the slope of the tangent to the curve at any point \( x \). In this case, the derivative \( f'(x) = 2x \) implies that as \( x \) increases, the slope increases linearly, meaning our mountain gets steeper and steeper.

• To differentiate a function, identify the type of function it is (e.g., polynomial, trigonometric, exponential) and apply the corresponding differentiation rules.
• Use these rules to transform your understanding of how inputs relate to outputs.

Logarithmic Differentiation

Logarithmic differentiation is a technique that becomes very handy when dealing with complex products or quotients. Logarithms have properties that simplify multiplication and division, making difficult differentiation problems much simpler. For instance, the logarithmic property \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \) is extremely useful.

In this type of differentiation, we take the natural logarithm of both sides of an equation, differentiating using the chain rule to simplify. This is particularly effective for functions like \( y = g(x)^{h(x)} \), where isolating variables for differentiation might be complex directly. Here's how it works:

• Apply logarithms to the function to break it down into simpler parts.
• Differentiate each part separately, making sure you apply the rules of derivatives we know.
• Combine the results to find the derivative of the original function.

Chain Rule

The chain rule is a powerful tool for differentiation that allows us to handle complex functions composed of other functions, such as \( f(g(x)) \). It's like peeling an onion layer by layer; you must differentiate each layer from the outside inwards.

When you encounter a composite function, the chain rule states that the derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically, if \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

• Identify the inner and outer functions within a given function.
• Differentiate the outer function while keeping the inner function intact.
• Multiply the result by the derivative of the inner function.

Using the chain rule, we can efficiently tackle challenging calculus problems, especially those with nested functions, ensuring that we peel back each layer of complexity effectively.