Problem 27 - Calculus and its applications

Question

Find \(y^{\prime \prime}\) $$ y=\left(x^{3}-x\right)^{3 / 4} $$

Step-by-Step Solution

Answer

The second derivative is \(y'' = \frac{3}{4} \left[ (3x^2 - 1) \left( -\frac{1}{4}(x^3-x)^{-5/4}(3x^2-1) \right) + (x^3 - x)^{-1/4}(6x) \right]\).

Step 1: Identify the Given Problem

The problem requires finding the second derivative, denoted as \(y''\), of the function \(y = (x^3 - x)^{3/4}\).

Step 2: Find the First Derivative (Chain Rule)

First, we apply the chain rule to differentiate \(y = (x^3 - x)^{3/4}\). Let \(u = x^3 - x\), so \(y = u^{3/4}\). The derivative of \(y\) with respect to \(u\) is \(\frac{dy}{du} = \frac{3}{4} u^{-1/4}\). The derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx} = 3x^2 - 1\). By applying the chain rule, we claim that \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \frac{3}{4} (x^3 - x)^{-1/4} (3x^2 - 1)\).

Step 3: Simplify the First Derivative

Simplify \(\frac{dy}{dx} = \frac{3}{4} (x^3 - x)^{-1/4} (3x^2 - 1)\) to \(y' = \frac{3}{4} (3x^2 - 1)(x^3 - x)^{-1/4}\).

Step 4: Find the Second Derivative

To find the second derivative \(y''\), apply the product rule to \(y' = \frac{3}{4} (3x^2 - 1)(x^3 - x)^{-1/4}\). Let \(u = 3x^2 - 1\) and \(v = (x^3 - x)^{-1/4}\). The derivatives are \(u' = 6x\) and \(v' = -\frac{1}{4}(x^3-x)^{-5/4}(3x^2-1)\). We apply the product rule as follows: $$y'' = \frac{3}{4} \left[ (3x^2 - 1) v' + (x^3 - x)^{-1/4} u' \right].$$\

Step 5: Combine Terms and Simplify

Substitute \(u'\) and \(v'\) back into the expression for \(y''\) to get: \(y'' = \frac{3}{4} \left[ (3x^2 - 1) \left( -\frac{1}{4}(x^3-x)^{-5/4}(3x^2-1) \right) + (x^3 - x)^{-1/4}(6x) \right]\). Simplify each term carefully to find the final expression for \(y''\).

Key Concepts

Understanding the Chain Rule

When you hear about the chain rule, think of it as a way to differentiate complex functions that "nest" layers inside one another. Imagine peeling an onion, layer by layer. Here, you have a function like \((x^3 - x)^{3/4}\) where one function is wrapped inside another. You need to differentiate the outer function and then multiply by the derivative of the inner function.

Here's how it works:

• Identify the outer function and the inner function. In our exercise, the outer function is \(u^{3/4}\) and the inner function is \(x^3 - x\).
• Differentiating the outer function with respect to the inner function involves dropping the power down and reducing it by one: \(\frac{3}{4}u^{-1/4}\).
• Differentiate the inner function normally, \(3x^2 - 1\).
• Multiply these derivatives to apply the chain rule: \(\frac{3}{4} (x^3 - x)^{-1/4} (3x^2 - 1)\).

By using the chain rule, you efficiently break down the complex problem into manageable steps.

Applying the Product Rule

The product rule is your tool for differentiating when you have two functions multiplying each other. Think of it as dividing the workload between two partners, each handling their part effectively. In this exercise, once you've applied the chain rule to find the first derivative \(y'\), the product rule comes in to differentiate a combination of two functions.

Here's an outline:

• Identify the two functions involved. For \(y' = \frac{3}{4} (3x^2 - 1)(x^3 - x)^{-1/4}\), they are \(u = 3x^2 - 1\) and \(v = (x^3 - x)^{-1/4}\).
• Compute the derivative of each function: \(u' = 6x\) and find \(v'\) using the chain rule again: \(v' = -\frac{1}{4}(x^3-x)^{-5/4}(3x^2-1)\).
• Plug these into the product rule formula: \(y'' = \frac{3}{4} \left[ (3x^2 - 1) v' + (x^3 - x)^{-1/4} u' \right]\).

The product rule ensures you account for changes in both functions as they vary with respect to \(x\).

Mastering Differentiation Steps

Differentiating might seem like a process with lots of rules, but it's really about breaking the problem down into bite-sized tasks. With the given function \(y = (x^3 - x)^{3/4}\), each step builds on the last.Here's the roadmap:

• Step 1: Chain Rule Application - Simplify by setting \(u = x^3 - x\), then use the chain rule to find the first derivative, \(y'\).
• Step 2: Simplification - Tidy up your solution, combining like terms and simplifying where possible.
• Step 3: Product Rule Application - Use for finding the second derivative of \(y' = \frac{3}{4} (3x^2 - 1)(x^3 - x)^{-1/4}\).
• Step 4: Solve and Simplify - Combine the differentiated components and simplify each term, calculating the second derivative \(y''\).

Whenever you differentiate, always write each step clearly. Check your work as you go along—it's a safeguard against simple mistakes. Remember, each step is another piece of the puzzle working towards a complete picture!