Problem 69 - Calculus and its applications

Question

Differentiate. $$ g(x)=x^{4} \ln |6 x| $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = 4x^3 \ln|6x| + x^3 \).

Step 1: Recognize the Product Rule

The function is a product of two differentiable functions: \( x^4 \) and \( \ln|6x| \). We will use the product rule, \( (uv)' = u'v + uv' \). Set \( u = x^4 \) and \( v = \ln|6x| \).

Step 2: Differentiate \( u = x^4 \)

Find the derivative of \( u = x^4 \). \[ u' = \frac{d}{dx}(x^4) = 4x^3 \]

Step 3: Differentiate \( v = \ln|6x| \)

Find the derivative of \( v = \ln|6x| \) using the chain rule. The derivative of \( \ln|x| \) is \( \frac{1}{x} \). \[ v' = \frac{d}{dx}(\ln|6x|) = \frac{1}{6x} \cdot 6 = \frac{1}{x} \]

Step 4: Apply the Product Rule

Substitute \( u, u', v, \) and \( v' \) into the product rule formula. \[ g'(x) = u'v + uv' = (4x^3)(\ln|6x|) + (x^4)(\frac{1}{x}) \]

Step 5: Simplify the Expression

Simplify the terms obtained from applying the product rule. \[ g'(x) = 4x^3 \ln|6x| + x^3 \] Combine like terms if possible.

Key Concepts

Product Rule

In calculus, the product rule is a fundamental tool for differentiating functions that are products of two or more differentiable functions. When you have a function composed of the product of two functions, say \( u(x) \) and \( v(x) \), the product rule helps you find the derivative of their product. The rule is expressed mathematically as:\[ (uv)' = u'v + uv' \]Here’s how it works in practice:

• Identify the two functions: First, identify the two differentiable functions that make up the product.
• Differentiate individually: Differentiate each function separately. That means find \( u' \) and \( v' \).
• Apply the formula: Substitute into the product rule formula. Multiply \( u' \) by \( v \), and \( u \) by \( v' \), and add them together.

This rule ensures you correctly account for the way both functions contribute to the change in the product.

Chain Rule

The chain rule is another essential differentiation tool. It is used whenever you need to differentiate a composite function—a function of a function. If you have a function \( y = f(g(x)) \), the chain rule provides a systematic way to find its derivative:\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]The chain rule works as follows:

• Identify the outer and inner functions: The function \( g(x) \) is the "inner" function and \( f \) is the "outer" function.
• Differentiate separately: First, find the derivative of the outer function with respect to the inner function, \( f'(g(x)) \). Then find the derivative of the inner function, \( g'(x) \).
• Multiply the derivatives: Multiply these two derivatives to get the total derivative of the composite function.

The chain rule is particularly powerful for handling logarithmic, exponential, and trigonometric functions that are nested within another function.

Logarithmic Differentiation

Logarithmic differentiation is a technique used for differentiating functions that are difficult to differentiate using standard rules, especially when the function is a product, quotient, or a power involving variables. It involves taking the natural logarithm on both sides of an equation \( y = f(x) \), and then differentiating to simplify the process:1. Apply logarithms: Start by taking the natural logarithm (\( \ln \)) of both sides of the equation \( y = f(x) \). This uses the property of logarithms to transform products into sums, which are easier to differentiate.2. Differentiate using the chain rule: Differentiate both sides with respect to \( x \). Utilize the chain rule to handle the \( \ln(y) \), keeping in mind that \( \frac{d}{dx} \ln(y) = \frac{1}{y} \cdot \frac{dy}{dx} \).3. Solve for \( \frac{dy}{dx} \): Rearrange your differentiated equation to solve for \( \frac{dy}{dx} \).This method simplifies differentiation considerably, particularly for functions where direct application of differentiation rules is cumbersome.