Problem 12 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\frac{\left(1-3 x^{2}\right)^{3}}{\left(3-x^{2}\right)^{2}} $$

Step-by-Step Solution

Answer

Differentiate using the quotient rule and chain rule; the derivative is complex.

Step 1: Identify the Differentiation Rule Needed

The function is given as a fraction of two functions, suggesting the quotient rule will be necessary. Each component of the fraction involves powers, indicating that the chain rule might also be applied.

Step 2: Apply the Quotient Rule

The quotient rule for derivatives is \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \), where \( u = (1-3x^2)^3 \) and \( v = (3-x^2)^2 \). Calculate \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) next.

Step 3: Differentiate the Numerator using the Chain Rule

Let \( u = (1-3x^2)^3 \). Apply the chain rule: \( \frac{du}{dx} = 3(1-3x^2)^2 \cdot \frac{d}{dx}(1-3x^2) = 3(1-3x^2)^2 \cdot (-6x) = -18x(1-3x^2)^2 \).

Step 4: Differentiate the Denominator using the Chain Rule

Let \( v = (3-x^2)^2 \). Again, use the chain rule: \( \frac{dv}{dx} = 2(3-x^2) \cdot \frac{d}{dx}(3-x^2) = 2(3-x^2) \cdot (-2x) = -4x(3-x^2) \).

Step 5: Combine Results in the Quotient Rule Formula

Substitute \( \frac{du}{dx} \) and \( \frac{dv}{dx} \) into the quotient rule: \[ \frac{d}{dx}f(x) = \frac{(3-x^2)^2(-18x(1-3x^2)^2) - (1-3x^2)^3(-4x(3-x^2))}{((3-x^2)^2)^2} \].

Step 6: Simplify the Expression

Distribute and simplify each term in the numerator: \[ -18x(3-x^2)^2(1-3x^2)^2 + 4x(1-3x^2)^3(3-x^2) \]. Then simplify the fraction, continuing with any algebraic reduction necessary to explore cancelation or factoring benefits.

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus differentiation that helps us determine the derivative of a function that is expressed as a quotient of two other functions. When a function is written in the form \( \frac{u(x)}{v(x)} \), the quotient rule states that its derivative is given by the formula:

• \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \)

This rule plays a critical role when dealing with complex fractions in differentiation. In our exercise, \( u = (1-3x^2)^3 \) and \( v = (3-x^2)^2 \). The quotient rule is applied to find \( \frac{d}{dx}f(x) \) by utilizing the derivatives of \( u \) and \( v \). Knowing how to apply the quotient rule allows us to differentiate many real-world problems where relationships are modeled by ratios or rates. When using this rule, the key steps are to first find the derivatives \( \frac{du}{dx} \) and \( \frac{dv}{dx} \), then plug these into the quotient rule formula. It's crucial to ensure that you square the denominator correctly to avoid errors. Breaking these steps helps reveal the derivative of the fraction clearly.

Chain Rule

The chain rule is an essential tool in calculus used when dealing with compositions of functions. It's particularly handy when you have a function inside another function, as seen in expressions like \( (1-3x^2)^3 \). The chain rule allows us to take the derivative of such composite functions. To find the derivative using the chain rule, follow these steps:

• Differentiate the outer function, keeping the inner function unchanged.
• Multiply this result by the derivative of the inner function.

In our exercise, for \( u = (1-3x^2)^3 \), we apply the chain rule:

• Differentiating the outside, \( 3(1-3x^2)^2 \), and then multiplying by the derivative of the inside, \(-6x\), gives \( \frac{du}{dx} = -18x(1-3x^2)^2 \).
• For \( v = (3-x^2)^2 \), a similar process yields \( \frac{dv}{dx} = -4x(3-x^2) \).

The chain rule simplifies the differentiation of complex composite functions, particularly when these functions involve polynomial expressions nested within each other.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. In calculus, particularly when applying rules like the quotient and chain rules, you often end up with complex fractions that are daunting at first glance. The goal is to simplify these results without losing the correct relationship or information of the expression. In our solution process, after applying the quotient rule, we reach the fractional expression: \[ \frac{-18x(3-x^2)^2(1-3x^2)^2 + 4x(1-3x^2)^3(3-x^2)}{((3-x^2)^2)^2} \] To simplify this, distribute and combine like terms in the numerator while looking for cancelation opportunities.

• Factor out common terms where possible to reduce the fraction.
• Carefully simplify the powers and coefficients.

Effective fraction simplification makes calculus problems manageable and prepares expressions for further analysis or for evaluating at specific points. Practice identifying common factors and applying algebraic techniques to see how the fraction can be made more compact, making it easier to interpret and use in subsequent calculations.