Problem 47 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ y=\log _{4}(5 x+1) $$

Step-by-Step Solution

Answer

The derivative of \( y=\log _{4}(5x+1) \) is \( y'= \frac{5}{(5x+1)\ln 4} \)

Step 1: Apply the Logarithmic Properties

To proceed, we could convert the base 4 logarithm into a natural logarithm using properties of logarithms. In this case, we use the change of base formula which states that \( \log _{a}b = \frac{\ln b}{\ln a} \). Thus, the function y becomes: \( y=\log _{4}(5x+1) = \frac{\ln (5x+1)}{\ln 4} \)

Step 2: Differentiate Using Chain Rule

After converting to natural logarithm, we can take the derivative of y using chain rule. The chain rule is a formula to compute the derivative of a composite function. When differentiating, the chain rule says that we differentiate the outside function first and then differentiate the inside function. Here, the outside function is the natural logarithm (whose derivative is 1/x) and the inside function is \( 5x+1 \). So, the derivative of y with respect to x is \( \frac{1}{5x+1} \times 5 - \frac{0}{(\ln 4)^2} = \frac{5}{5x+1} \times \frac{1}{\ln 4} \)

Step 3: Simplify the Derivative

We got the derivative but it is a complex fraction and can be simplified. The final derivative of the function y with respect to x is \( \frac{5}{5x+1} \times \frac{1}{\ln 4} = \frac{5}{(5x+1)\ln 4} \)

Key Concepts

Change of Base Formula

When dealing with logarithms, you may encounter bases that are not convenient to work with directly. This is where the change of base formula comes in handy. The formula allows you to express a logarithm in terms of logarithms with a different base, typically the natural logarithm (ln) or the common logarithm (base 10, often denoted as 'log'). This transformation can simplify calculations, especially in calculus.
For instance, the change of base formula states that for any positive numbers \( a \), \( b \), and \( x \) (where \( a eq 1 \) and \( b eq 1 \)), the equation is given by:

• \( \log_{a}(x) = \frac{\ln(x)}{\ln(a)} \)

This formula is incredibly useful in differentiation problems, as converting to natural logarithms (ln) can simplify the differentiation process. In the original problem, \( \log_{4}(5x+1) \) is converted to \( \frac{\ln(5x+1)}{\ln(4)} \), making it easier to work with the derivative using the properties of natural logarithms.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions, i.e., functions within functions. It's a way to unpack the intricate layers of a mathematical expression by focusing on the outer function first, then moving inward. Understanding this rule is crucial for tackling complex derivatives.
To recall, the chain rule states that if you have a composite function \( f(g(x)) \), its derivative is given by:

• First, find the derivative of the outer function \( f \), leaving the inner function \( g(x) \) unchanged.
• Then, multiply this result by the derivative of the inner function \( g(x) \).

In practical terms, for the function \( y = \ln(5x+1) \), you have:

• Outer function: \( \ln(u) \), derivative \( \frac{1}{u} \)
• Inner function: \( 5x + 1 \), derivative \( 5 \)
• Therefore, the derivative of \( y \) is the product of these,\( \frac{1}{5x+1} \times 5 \), which further multiplies by \( \frac{1}{\ln(4)} \), due to the change of base.

This demonstrates how the independence of each function's derivative component works together to reveal the complexity underneath the surface with ease.

Logarithmic Properties

Understanding the properties of logarithms can greatly simplify differentiation tasks, as they provide tools for manipulation before applying calculus techniques. Logarithmic properties are related rules that help to combine, break down, and simplify logarithmic terms.

This includes pasting together or tearing apart logarithms using the following identities:
• Product rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
• Power rule: \( \log_b(m^n) = n \cdot \log_b(m) \)

These properties allow you to restructure and thus simplify expressions, preparing them for smooth differentiation. In our exercise, converting the expression through the change of base allowed the function \( y = \log_{4}(5x+1) \) to switch over to the more differentiation-friendly ln format using \( \frac{\ln(5x+1)}{\ln(4)} \). This adjusted form made the utilization of the chain rule straightforward, finally simplifying the derivative down to \( \frac{5}{(5x+1)\ln 4} \). The step-by-step techniques, with the aid of these properties, unravel the complexity and make finding derivatives manageable.