Problem 26 - Calculus and its applications

Question

Differentiate. $$ y=6^{x} \cdot \log _{7} x $$

Step-by-Step Solution

Answer

The derivative is \( y' = 6^{x} \cdot \ln(6) \cdot \log_{7} x + \frac{6^{x}}{x \cdot \ln 7} \).

Step 1: Apply Product Rule

To differentiate the function \( y = 6^{x} \cdot \log_{7} x \), we need to use the product rule. The product rule states that if \( y = u \cdot v \), then \( y' = u' \cdot v + u \cdot v' \). Here, \( u = 6^{x} \) and \( v = \log_{7} x \).

Step 2: Differentiate \( u = 6^{x} \)

To differentiate \( 6^{x} \), use the formula for differentiating exponential functions: \( \frac{d}{dx} a^{x} = a^{x} \cdot \ln(a) \). Hence, \( \frac{d}{dx} 6^{x} = 6^{x} \cdot \ln(6) \).

Step 3: Differentiate \( v = \log_{7} x \)

To differentiate \( \log_{7} x \), first express it in terms of the natural logarithm: \( \log_{7} x = \frac{\ln x}{\ln 7} \). Using the formula \( \frac{d}{dx} \ln x = \frac{1}{x} \), we get \( \frac{d}{dx} \log_{7} x = \frac{1}{x \cdot \ln 7} \).

Step 4: Substitute into the Product Rule

Replace \( u' \), \( u \), \( v' \), and \( v \) in the product rule formula. So, \( y' = (6^{x} \cdot \ln(6)) \cdot \log_{7} x + 6^{x} \cdot (\frac{1}{x \cdot \ln 7}) \).

Step 5: Simplify the Expression

Combine and simplify the expression: \[ y' = (6^{x} \cdot \ln(6) \cdot \log_{7} x) + \left( \frac{6^{x}}{x \cdot \ln 7} \right) \]. This is the derivative of the original function.

Key Concepts

Product Rule

Let's begin with the product rule, an essential tool in differential calculus. When you have a function that is a product of two other functions, such as \( y = u \cdot v \), the derivative is not as straightforward as taking the derivative of each separately. Instead, we apply the product rule. The product rule states:

• If \( y = u \cdot v \), then the derivative \( y' = u' \cdot v + u \cdot v' \).

This can be summarized as "derivative of the first times the second plus the first times the derivative of the second." If you remember only this structure, you will handle many differentiation problems with ease. By using the product rule, we ensure that both parts of a product function are properly accounted for when finding the derivative. It's particularly helpful when differentiating complex functions that can't be simplified into sums or differences.

Exponential Functions

Exponential functions are powerful and are of the form \( a^x \), where \( a \) is a constant and \( x \) is an exponent. These functions grow very rapidly and their differentiation involves a special rule. The derivative of an exponential function \( a^x \) is given by: \[\frac{d}{dx} a^x = a^x \cdot \ln(a)\] Here, \( \ln(a) \) is the natural logarithm of the base \( a \).

• For instance, for \( 6^x \), the derivative is \( 6^x \cdot \ln(6) \).
• This formula is derived from the fact that exponential functions are closely tied to their own rates of growth.

Exponential functions are not only found in pure math problems but also in real-world applications like population growth, radioactive decay, and financial calculations. Understanding these proportions makes applying their derivatives intuitive.

Logarithmic Differentiation

Logarithmic differentiation is often used to differentiate complicated functions that involve logarithms. One special case is when dealing with functions like \( \log_{b} x \), which can be tricky without leveraging natural logarithms (\( \ln \)). For the differentiation of \( \log_{b} x \), it's useful to rewrite it as \( \frac{\ln x}{\ln b} \). Then, using the differentiation rule for natural logs: \[\frac{d}{dx} \ln x = \frac{1}{x}\] Applying this gives: \[\frac{d}{dx} \log_{b} x = \frac{1}{x \cdot \ln b}\]

• This approach effectively simplifies the process of differentiating with respect to \( x \).
• It's particularly useful when the logarithmic base is other than \( e \), the base of natural logarithms.

Logarithmic differentiation offers a streamlined method to deal with otherwise complicated derivatives, especially when combined with other differentiation rules such as the product or chain rules.