Problem 70 - Calculus for Biology and Medicine

Question

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{3 / x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = x^{3/x} \left( -\frac{3}{x^2} \ln(x) + \frac{3}{x^2} \right) \).

Step 1: Express the Function in Logarithmic Form

First, take the natural logarithm of both sides of the equation. So, \[ y = x^{3/x} \] becomes\[ \ln(y) = \ln(x^{3/x}) \].Using the properties of logarithms, you can rewrite the right-hand side as \[ \ln(y) = \frac{3}{x} \ln(x) \].

Step 2: Differentiate Both Sides Implicitly

Differentiate both sides with respect to \( x \). The derivative of \( \ln(y) \) is \( \frac{1}{y} \frac{dy}{dx} \). For the right-hand side, use the product rule where \( u = \frac{3}{x} \) and \( v = \ln(x) \).So, \[ \frac{1}{y} \frac{dy}{dx} = \left( -\frac{3}{x^2} \ln(x) + \frac{3}{x} \cdot \frac{1}{x} \right). \]

Step 3: Simplify the Expression

Simplify the derivative expression further:\[ \frac{1}{y} \frac{dy}{dx} = -\frac{3}{x^2} \ln(x) + \frac{3}{x^2} \].This is the simplified form of the right-hand side.

Step 4: Solve for \( \frac{dy}{dx} \)

Multiply both sides by \( y \) to solve for the derivative:\[ \frac{dy}{dx} = y \left( -\frac{3}{x^2} \ln(x) + \frac{3}{x^2} \right). \]

Step 5: Substitute Back the Original Function

Recall \( y = x^{3/x} \). Substitute back to express \( \frac{dy}{dx} \) in terms of \( x \):\[ \frac{dy}{dx} = x^{3/x} \left( -\frac{3}{x^2} \ln(x) + \frac{3}{x^2} \right). \]

Key Concepts

Understanding Derivatives

The concept of the derivative is central in calculus. It measures how a function changes as its input changes. Think of it as finding the rate at which one quantity changes relative to another. For a function \( f(x) \), the derivative, denoted as \( f'(x) \) or \( \frac{df}{dx} \), tells us how the function \( f \) changes with respect to \( x \). In our example, we want to find the derivative of \( f(x) = x^{3/x} \) using logarithmic differentiation.

Logarithmic differentiation is particularly useful when dealing with complicated functions such as products, quotients, or exponents like \( x^{3/x} \). It simplifies the process by taking the natural logarithm of the function, which makes deriving easier using the properties of logarithms. By using these properties, such as transforming powers into products, functions become more manageable. This aids in smooth and simple differentiation.

Implicit Differentiation Explained

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of the others. Instead, the equation is composed of two or more variables intermixed, similarly to the example \( y = x^{3/x} \). With implicit differentiation, both sides are differentiated with respect to a variable, often \( x \), while treating \( y \) as an implicit function of \( x \).

In our exercise, after using logarithmic differentiation, we reach a step where we need to differentiate \( \ln(y) = \frac{3}{x} \ln(x) \). By applying implicit differentiation, we need to recognize that \( y \) is a function of \( x \). Thus when differentiating \( \ln(y) \) with respect to \( x \), we get \( \frac{1}{y} \cdot \frac{dy}{dx} \).

This technique is advantageous in finding derivatives of complicated functions because it allows us to solve for \( \frac{dy}{dx} \) directly, using the relationship of \( y \) in terms of \( x \). It's also very helpful when dealing with equations where direct differentiation isn't feasible.

Implementation of the Product Rule

The product rule is a foundational differentiation rule that is used to find the derivative of a product of two functions. When you have a function \( u(x) \) multiplied by another function \( v(x) \), the product rule states that the derivative of \( u(x) \cdot v(x) \) is \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \).

In the logarithmic differentiation of \( x^{3/x} \), the expression \( \frac{3}{x} \ln(x) \) is differentiated using the product rule in Step 2 of our solution. Here, \( u = \frac{3}{x} \) and \( v = \ln(x) \). To find the derivative of their product, we need to calculate the derivatives of each individually—\( u'(x) = -\frac{3}{x^2} \) and \( v'(x) = \frac{1}{x} \)—and then apply the product rule.

This approach breaks down complex derivatives into simpler, manageable steps. By applying the product rule we ensure that each component of the function is differentiated accurately, making the overall process of finding complicated derivatives much simpler and organized.