Problem 69 - Calculus and its applications

Question

Use the Chain Rule to differentiate each function. You may need to apply the rule more than once. $$ f(x)=\sqrt{x^{2}+\sqrt{1-3 x}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{1}{2}(x^2 + \sqrt{1 - 3x})^{-1/2} \cdot \left( 2x - \frac{3}{2}(1 - 3x)^{-1/2} \right). \)

Step 1: Identify the Outer Function

The given function is \[ f(x) = \sqrt{x^2 + \sqrt{1 - 3x}}. \]Let's rewrite the square root as a power to identify the outermost function:\[ f(x) = (x^2 + \sqrt{1 - 3x})^{1/2}. \]The outer function is the square root or equivalently, raising to the power of \( \frac{1}{2} \).

Step 2: Apply the Chain Rule to the Outer Function

Differentiate the outer function \( (u)^{1/2} \) where \( u = x^2 + \sqrt{1 - 3x} \). The derivative with respect to \( u \) is\[ \frac{1}{2} u^{-1/2}. \]By the chain rule, we multiply by the derivative of the inner function \( u \) with respect to \( x \):\[ f'(x) = \frac{1}{2} (x^2 + \sqrt{1 - 3x})^{-1/2} \cdot \frac{d}{dx}(x^2 + \sqrt{1 - 3x}). \]

Step 3: Differentiate the Inner Function

The inner function is \( u = x^2 + \sqrt{1 - 3x} \). We need to find \( \frac{d}{dx}(u) \): - The derivative of \( x^2 \) is \( 2x \).- The derivative of \( \sqrt{1 - 3x} \) can be found using the chain rule. Rewrite \( \sqrt{1 - 3x} = (1 - 3x)^{1/2} \).

Step 4: Apply the Chain Rule to Nested Function

For \( (1 - 3x)^{1/2} \), the outer function is \( (v)^{1/2} \) and the inner function is \( v = 1 - 3x \). Differentiate \( (v)^{1/2} \) with respect to \( v \):\[ \frac{1}{2}(1 - 3x)^{-1/2}. \]And then multiply by the derivative of \( v = 1 - 3x \):\[ \frac{d}{dx}(1 - 3x) = -3. \]Thus, the derivative is\[ \frac{1}{2}(1 - 3x)^{-1/2} \cdot (-3) = -\frac{3}{2}(1 - 3x)^{-1/2}. \]

Step 5: Combine Derivatives

Substitute back the result from Step 4 into the derivative of \( u \):\[ \frac{d}{dx}(x^2 + \sqrt{1 - 3x}) = 2x - \frac{3}{2}(1 - 3x)^{-1/2}. \]Now, substitute this expression back into the equation from Step 2:\[ f'(x) = \frac{1}{2}(x^2 + \sqrt{1 - 3x})^{-1/2} \cdot \left( 2x - \frac{3}{2}(1 - 3x)^{-1/2} \right). \]

Key Concepts

Differentiation

Differentiation is a core concept in calculus that focuses on determining the derivative of a function, which captures how the function changes as its input changes. Imagine tracking the speed of a car on a highway; just as that speed can change, functions can vary as well. By finding the derivative, we can understand this rate of change more precisely.
In the given exercise, we use differentiation to uncover how the function \[ f(x) = \sqrt{x^2 + \sqrt{1 - 3x}} \] responds to changes in the variable \( x \). This process helps us determine not just the speed of change but exactly how swiftly different parts of our function respond when \( x \) changes.

• The primary tool for differentiation in this scenario is the Chain Rule, a method allowing us to handle functions composed of multiple layers.
• Breaking down these layers, as we'll see, involves isolating each effect and studying it individually.
• This is crucial because solving complex differentiation problems without a structured technique would be tricky and error-prone.

Differentiating step by step aligns with how each part of a composition contributes to the whole change—a foundational lesson in understanding intricate functions.

Outer Function

Looking at a complex function, it's often layered, much like an onion. The outer function is what wraps around everything else, defining the broadest part of our operation. In our exercise, this outer layer is identified as the square root: \[ f(x) = \sqrt{x^2 + \sqrt{1 - 3x}} \] or equivalently, \[ (x^2 + \sqrt{1 - 3x})^{1/2}. \]

• The outer function here is raising to the power of \( 1/2 \). This viewpoint allows us to apply specific rules to differentiate it.
• In differentiation, handling the outer function gets us started as it dictates the initial flow of derivative operations.
• This means that we differ it to see how the entire function's outer shell adjusts with inputs.

By addressing the outer function first, we set the scene for how changes in its inner functions, its ingredients, will ultimately fit into the broader picture of change.

Inner Function

Just below the surface of our problem lies the inner function, which provides the structure around which the outer function wraps itself. In our example, this inner function is: \[ u = x^2 + \sqrt{1 - 3x}. \]

• Successfully identifying and differentiating the inner function is pivotal because it carries complexity that affects the overall rate of change.
• Here, we have combined terms, including another square root. This necessitates a second application of the Chain Rule.
• The inner functions, like \( x^2 \) and \( \sqrt{1 - 3x} \), each contribute a unique adjustment path, requiring us to individually handle and derive them before forming a composite result.

Working through inner layers ensures that when they are "unwrapped," the changes from each segment interact correctly with the overall operation. Hence, understanding and managing the inner functions provides completeness to our differentiation process.