Problem 44 - Calculus for Biology and Medicine

Question

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\frac{x}{2\left(x^{2}-1\right)^{2}-1}\right)^{2}\)

Step-by-Step Solution

Answer

Use the chain and quotient rules to find \( \frac{dy}{dx} \) and simplify the expression.

Step 1: Identify Outer Function

The first function to differentiate is the outer one: \[ y = u^2 \] where \( u = \frac{x}{2(x^2 - 1)^2 - 1} \). We will apply the power rule first.

Step 2: Differentiate Outer Function

Apply the power rule to differentiate \( y = u^2 \). \[ \frac{dy}{du} = 2u \]

Step 3: Identify Intermediate Function

The intermediate function \( u = \frac{x}{2(x^2 - 1)^2 - 1} \) needs to be differentiated using the quotient rule. Define \( f(x) = x \) and \( g(x) = 2(x^2 - 1)^2 - 1 \) for the quotient \( u = \frac{f(x)}{g(x)} \).

Step 4: Differentiate Intermediate Function

Apply the quotient rule which states \[ \frac{d}{dx} \left( \frac{f}{g} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \] to find \( \frac{du}{dx} \). * \( f'(x) = 1 \).* To find \( g'(x) \), apply the chain rule to \( g(x) = 2(x^2 - 1)^2 - 1 \).

Step 5: Differentiate Inside Function

First differentiate \((x^2 - 1)^2\) using the chain rule. Let \( h(x) = x^2 - 1 \), so \( (h(x))^2 \) is differentiated as\[ (h(x))^2 = 2h(x)h'(x) = 2(x^2 - 1) \cdot 2x = 4x(x^2 - 1) \]. Thus, \[ g'(x) = 2 \cdot 4x(x^2 - 1) = 8x(x^2 - 1) \].

Step 6: Apply Quotient Rule with Results

Substitute \( f(x) = x \), \( f'(x) = 1 \), \( g(x) = 2(x^2 - 1)^2 - 1 \), and \( g'(x) = 8x(x^2 - 1) \) into the quotient rule: \[ \frac{du}{dx} = \frac{1 \cdot [2(x^2 - 1)^2 - 1] - x \cdot 8x(x^2 - 1)}{([2(x^2 - 1)^2 - 1])^2} \].

Step 7: Calculate Final Derivative

Combine the derivatives using the chain rule. We have \( \frac{dy}{du} = 2u \) and \( \frac{du}{dx} \).Thus, \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2 \cdot \frac{x}{2(x^2 - 1)^2 - 1} \cdot \frac{1 \cdot [2(x^2 - 1)^2 - 1] - x \cdot 8x(x^2 - 1)}{([2(x^2 - 1)^2 - 1])^2} \].

Step 8: Simplify the Derivative Expression

Simplify the expression \[ \frac{dy}{dx} = \frac{2x \cdot [2(x^2 - 1)^2 - 1] - 8x^2(x^2 - 1)}{([2(x^2 - 1)^2 - 1])^3} \].Cancel, expand and simplify terms as needed.

Key Concepts

Quotient Rule

Differentiating a function that is expressed as a quotient involves using the Quotient Rule. This rule is essential when dealing with expressions where one function is divided by another. Imagine you have two functions: \( f(x) \) and \( g(x) \), and you need to find the derivative of \( \frac{f(x)}{g(x)} \). The Quotient Rule provides a formula for this:

• If \( u = \frac{f(x)}{g(x)} \), then \( \frac{du}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \).

The formula might look intimidating, but it's quite straightforward:

• Start by finding the derivative of the numerator \( f'(x) \).
• Find the derivative of the denominator \( g'(x) \).
• Finally, plug these into the formula to get the derivative of the quotient.

Remember, it's crucial to correctly identify the parts of the fraction because this will guide the differentiation process. This approach ensures you find the derivative accurately, maintaining the division between the two functions.

Power Rule

The Power Rule is one of the fundamental tools for differentiating functions, especially when dealing with exponents. It helps simplify the derivatives of polynomial expressions. If you have a function of the form \( y = u^n \), the Power Rule states that the derivative \( \frac{dy}{du} \) will be \( nu^{n-1} \).

• Identify the base function, \( u \), and the exponent, \( n \).
• Use the rule to bring down the exponent as a coefficient (multiply), and subtract one from the original exponent.

For instance, differentiating \( y = u^2 \) would yield \( \frac{dy}{du} = 2u \). It's straightforward and comes in handy when you encounter composite functions where each component requires a different technique to differentiate. Always keep an eye out for functions structured this way, as they make differentiation much more efficient.

Derivative Simplification

After finding a derivative, either by using rules like the Chain Rule, the Quotient Rule, or the Power Rule, the next critical phase is simplifying the expression. Simplification involves reducing the derivative expression to its most compact form.

• Start by expanding any terms that can be simplified through basic algebra (like combining like terms).
• Look for opportunities to cancel out terms that appear in both the numerator and the denominator of the expression.

This step is crucial because a simplified derivative is easier to interpret and often critical for further analysis, like evaluating at specific points or solving equations. Simplification may involve factoring polynomials or even re-expressing radicals and complex fractions. Always double-check that you've simplified as fully as possible without altering the expression's meaning.