Problem 340 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\ln \left(4 x^{3}+x\right)$$

Step-by-Step Solution

Answer

\( f'(x) = \frac{12x^2 + 1}{4x^3 + x} \)

Step 1: Identify the function type

The function given is a logarithm of a composite function: \( f(x) = \ln(4x^3 + x) \). This requires the use of the chain rule for differentiation.

Step 2: Differentiate the outer function

The outer function is the natural logarithm \( \ln(u) \), where \( u = 4x^3 + x \). The derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).

Step 3: Differentiate the inner function

The inner function \( u \) is \( 4x^3 + x \). Differentiate it with respect to \( x \). The derivative is \( \frac{d}{dx}(4x^3 + x) = 12x^2 + 1 \).

Step 4: Apply the chain rule

The chain rule states that if you have a composite function \( f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \). Applying this, we get \( f'(x) = \frac{1}{4x^3 + x} \cdot (12x^2 + 1) \).

Step 5: Simplify the derivative

Multiply the derivatives obtained from Step 2 and Step 3. Hence, the derivative of the function \( f(x) \) is \( f'(x) = \frac{12x^2 + 1}{4x^3 + x} \).

Key Concepts

Chain Rule

The chain rule is an essential technique in calculus for differentiating composite functions. A composite function is a function made up of two or more functions, where one function is inside another. In simpler terms, it's like a set of Russian nesting dolls, where each doll is within another. To use the chain rule, you must differentiate the inner function and the outer function separately.
For example, if you have a function like \( f(g(x)) \), the chain rule states that you differentiate by multiplying the derivative of \( f \) (with respect to \( g(x) \)) by the derivative of \( g(x) \) (with respect to \( x \)). In math terms, this means \( f'(g(x)) \cdot g'(x) \).
This rule helps break down complex functions into manageable parts, allowing you to solve problems that otherwise seem daunting. It is particularly useful when dealing with logarithmic, exponential, and trigonometric functions nested inside each other.

Composite Function

When discussing composite functions, we are referring to a function created when one function is applied to the result of another, like \( f(g(x)) \). The input \( x \) first goes through \( g(x) \), and then its output goes through \( f(x) \). This structure leads to functions that can be a little more complex to work with.
The structure \( f \big(g(x)\big) \) ensures that you must consider both the outputs at each stage, making differentiation slightly more challenging. Nevertheless, with the correct approach, such as the chain rule, the process becomes manageable.

• First, identify the inner function, say \( g(x) \).
• Next, identify the outer function, say \( f(u) \), where \( u = g(x) \).
• Finally, differentiate both functions separately and apply the chain rule.

Using the chain rule is a key method for dealing with composite functions in differentiation, making this type of function easier to work with.

Natural Logarithm

The natural logarithm function, often denoted as \( \ln(x) \), is a logarithm with a base of \( e \), where \( e \approx 2.718 \). It's a special function used extensively in calculus because of its unique properties. One of the most crucial properties is how its derivative behaves.
When you have a natural logarithm of a function, say \( \ln(u) \), the derivative with respect to \( u \) is \( \frac{1}{u} \). This makes it easier to work within various calculus problems, especially in conjunction with the chain rule.
Logarithmic differentiation also assists with multiplicative functions and can simplify problems involving exponents or powers. Understanding \( \ln(x) \) and its derivative is essential in calculus, allowing for solving integrals and differential equations effectively and efficiently. When it comes to solving these equations, knowing how to differentiate \( \ln(x) \) becomes an invaluable skill.