Problem 12 - Calculus with Concepts in Calculus

Question

Compute \(\partial z / \partial r\) and \(\partial z / \partial s\). $$ z=\ln u+\ln v ; u=4^{r s}, v=4^{r / s} $$

Step-by-Step Solution

Answer

\( \frac{\partial z}{\partial r} = (s + 1/s) \ln 4, \ \frac{\partial z}{\partial s} = (r - r/s^2) \ln 4.\)

Step 1: Express z in terms of u and v

The function is given as \( z = \ln u + \ln v \). This expression can be rewritten using the properties of logarithms: \( z = \ln (u \cdot v) \).

Step 2: Substitute expressions for u and v

We know that \( u = 4^{rs} \) and \( v = 4^{r/s} \). Therefore, the expression for \( z \) can be rewritten as follows: \( z = \ln (4^{rs} \cdot 4^{r/s}) \).

Step 3: Simplify the expression for z

Use the property of exponents that states that \( a^m \cdot a^n = a^{m+n} \). Thus, \( z = \ln(4^{rs + r/s}) = (rs + r/s)\ln 4 \).

Step 4: Differentiate z with respect to r

To find \( \frac{\partial z}{\partial r} \), use the chain rule: \( \frac{\partial z}{\partial r} = (s + 1/s) \ln 4 \).

Step 5: Differentiate z with respect to s

To find \( \frac{\partial z}{\partial s} \), use the chain rule: \( \frac{\partial z}{\partial s} = (r - r/s^2) \ln 4 \).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used in calculus for differentiating functions by employing the natural logarithm function. This method simplifies the process of taking derivatives, especially for functions that involve products, quotients, or are raised to complex powers. By taking the logarithm of both sides of an equation, complex expressions become easier to handle. Often, it is applied when direct differentiation is difficult or cumbersome. When we take the natural logarithm of a product, like in our original exercise with function \( z = \ln u + \ln v \), we can simplify this to \( z = \ln (u \cdot v) \).

• Advantages: Simplifies differentiation of complicated functions, especially those involving products or quotients.
• Process: Take the log of both sides, simplify using log properties, and then differentiate.

In the exercise, logarithmic differentiation is used to simplify the function involving powers of 4, making it easier to differentiate with respect to \( r \) and \( s \). This allows the application of the chain rule effectively.

Chain Rule in Calculus

The Chain Rule is a fundamental differentiation rule in calculus that is used to differentiate compositions of functions. It's particularly useful when dealing with nested functions where one function is inside another. The chain rule formula is generally expressed as:\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]This means you differentiate the outer function, evaluated at the inner function, and then multiply by the derivative of the inner function.
In the context of partial differentiation, such as in our exercise, the chain rule helps to find the derivative of \( z \) with respect to \( r \) and \( s \) by treating each variable independently while keeping others constant. For instance:

• With respect to \( r \): Differentiate \( (rs + r/s)\ln 4 \) as \( s + 1/s \).
• With respect to \( s \): Differentiate \( (rs + r/s)\ln 4 \) as \( r - r/s^2 \).

This rule is crucial for dealing with functions represented as compositions, enabling the breaking down of complex derivatives into simpler, more manageable parts.

Exponent Rules

Understanding exponent rules is essential for working with expressions that involve powers. In calculus, exponent rules simplify differentiation involving exponential expressions. The basic exponent rules include:

• \( a^m \cdot a^n = a^{m+n} \)
• \( (a^m)^n = a^{m \cdot n} \)
• \( a^{-m} = \frac{1}{a^m} \)

In the given problem, exponent rules are used to transform the product of powers into a single expression. Specifically, \( z = \ln(4^{rs} \cdot 4^{r/s}) \) is simplified by applying the rule \( a^m \cdot a^n = a^{m+n} \) to become \( z = \ln(4^{rs+r/s}) \). This simplification makes subsequent differentiation steps more straightforward.
These rules are not only key to simplifying expressions but are also fundamental when applying logarithmic differentiation, as they allow further simplification needed before applying the chain rule to find partial derivatives.