Problem 17 - Basic Engineering Mathematics

Question

Differentiate \(y=5 \ln 3 x\)

Step-by-Step Solution

Answer

The derivative is \( \frac{5}{x} \).

Step 1: Understand the Function

The given function is \(y = 5 \ln(3x)\). We need to differentiate this function with respect to \(x\). The function is composed of a constant multiple (5) and a natural logarithm function.

Step 2: Use the Chain Rule

The function can be rewritten as \( y = 5 \ln u \) where \( u = 3x \). According to the chain rule, the derivative of \(y\) with respect to \(x\) is the derivative of \(y\) with respect to \(u\) times the derivative of \(u\) with respect to \(x\).

Step 3: Differentiate \(y = 5 \ln u\) with Respect to \(u\)

The derivative of \( \ln u \) with respect to \( u \) is \( \frac{1}{u} \). Thus, the derivative of \( 5 \ln u \) with respect to \( u \) is \( 5 \cdot \frac{1}{u} = \frac{5}{u} \).

Step 4: Differentiate \(u = 3x\) with Respect to \(x\)

The derivative of \( u = 3x \) with respect to \( x \) is simply \( 3 \), since the derivative of \(a x\) is \(a\).

Step 5: Combine the Derivatives Using the Chain Rule

Applying the chain rule, we multiply the derivatives obtained in Step 3 and Step 4: \( \frac{d}{dx} y = \frac{5}{u} \cdot 3 = \frac{5}{3x} \cdot 3 = \frac{15}{3x} = \frac{5}{x} \). Therefore, the derivative is \( \frac{5}{x} \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function changes at any given point. This process is known as finding the derivative of the function. When we differentiate a function like the natural logarithm, we can determine how steep or flat the curve is at any point. This tells us how sensitive the function is to changes in its input.

Key ideas when differentiating:

• Understand the function: Break it down into simpler parts if possible.
• Determine if any basic differentiation rules apply: Such as constant, power, or more complex rules like the chain rule.
• Apply these rules carefully: Follow each step to ensure accuracy, especially when dealing with composite functions.

Chain Rule

The chain rule is a crucial technique in differentiation when dealing with composite functions. It allows us to differentiate a function that is nested within another function. For example, when differentiating a function like \(y = 5 \ln(3x)\), we use the chain rule because there are multiple layers - the outer logarithmic function and the inner linear function.

How the chain rule works:

• Identify the inner function \(u\) and the outer function \(f(u)\).
• First, differentiate the outer function with respect to \(u\).
• Then, differentiate the inner function with respect to the variable \(x\).
• Multiply the derivatives from the previous steps to get the final derivative.

This allows you to tackle complex differentiation tasks by breaking them down into simpler parts.

Natural Logarithm

A natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. The natural logarithm function is one of the most important functions in calculus due to its unique rate of growth and its appearance in various natural phenomena.

Differentiating the natural logarithm:

• The derivative of \(\ln(u)\) with respect to \(u\) is \(\frac{1}{u}\).
• This property allows you to easily find how quickly \(\ln(u)\) changes as \(u\) varies.

In our specific problem, differentiating \(5 \ln(3x)\) involves applying this property, along with using the chain rule, to handle the composite nature of the function.