Problem 11 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{3 x-1}{\sqrt{2 x^{2}-1}}\)

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{-3 + 2x}{(2x^2 - 1)\sqrt{2x^2 - 1}} \).

Step 1: Identify the Differentiation Rule

The function is in the form of a quotient \( \frac{u}{v} \), where \( u = 3x - 1 \) and \( v = \sqrt{2x^2 - 1} \). To differentiate this, we will use the quotient rule, which states: \( \left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2} \).

Step 2: Differentiate the Numerator

The numerator of the function is \( u = 3x - 1 \). The derivative of \( u \) with respect to \( x \) is \( u' = 3 \).

Step 3: Differentiate the Denominator

The denominator is \( v = \sqrt{2x^2 - 1} \). First, rewrite it as \( (2x^2 - 1)^{1/2} \). Using the chain rule, the derivative \( v' \) is \( \frac{1}{2}(2x^2 - 1)^{-1/2} \times 4x = \frac{2x}{\sqrt{2x^2 - 1}} \).

Step 4: Apply the Quotient Rule

Substitute the derivatives obtained into the quotient rule: \[ \left(\frac{3x - 1}{\sqrt{2x^2 - 1}}\right)' = \frac{\sqrt{2x^2 - 1}(3) - (3x - 1)\left(\frac{2x}{\sqrt{2x^2-1}}\right)}{(2x^2 - 1)} \].

Step 5: Simplify the Expression

After substituting the derivatives, simplify the expression: \[ \frac{3\sqrt{2x^2 - 1} - (3x - 1)\left(\frac{2x}{\sqrt{2x^2-1}}\right)}{2x^2 - 1} \]. Simplify further to obtain: \[ \frac{3(2x^2 - 1) - 2x(3x - 1)}{(2x^2 - 1)\sqrt{2x^2 - 1}} \].

Step 6: Final Simplification

Simplify the numerator: \( 6x^2 - 3 - (6x^2 - 2x) = -3 + 2x \). Therefore, the simplified derivative is \[ \frac{-3 + 2x}{(2x^2 - 1)\sqrt{2x^2 - 1}} \].

Key Concepts

Quotient Rule in Differentiation

The quotient rule is a fundamental tool for differentiating functions that are expressed as fractions, or quotients. If you have a function that can be written as \( \frac{u}{v} \), where both \( u \) and \( v \) are differentiable functions of \( x \), the quotient rule helps find the derivative. The formula is:

• \( \left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2} \)

This ensures that both the numerator and the denominator are considered, as they both play integral roles in the behavior of the function.
To apply the quotient rule:

• Identify the numerator \( u \) and denominator \( v \).
• Differentiate \( u \) to find \( u' \).
• Differentiate \( v \) to find \( v' \).
• Substitute \( u, v, u', \) and \( v' \) into the quotient rule formula.

This method allows you to differentiate complex rational functions with more ease, providing a clearer path to simplification and interpretation of derivatives.

Chain Rule for Complex Functions

The chain rule is an essential technique in calculus for taking the derivative of composite functions, functions formed by nesting one function inside another. It is stated as follows: if you have \( y = f(g(x)) \), where both \( f \) and \( g \) are differentiable functions, the derivative \( y' \) is given by:

• \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \)

This means you differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.
In our example, when differentiating the denominator \( v = \sqrt{2x^2 - 1} \), rewrite it as \( (2x^2 - 1)^{1/2} \). The chain rule allows us to differentiate it step-by-step:

• First, treat \( 2x^2 - 1 \) as the inside function, and \( (2x^2 - 1)^{1/2} \) as the outside function.
• Differentiate \( (2x^2 - 1)^{1/2} \) with respect to \( (2x^2 - 1) \).
• Then multiply by the derivative of \( 2x^2 - 1 \).

This simplifies the process, making it possible to find the derivative of more complex expressions efficiently.

Derivative Simplification

After deriving an expression, simplifying the derivative is crucial for ease of interpretation and application. Simplification usually involves:

• Combining like terms.
• Simplifying fractions.
• Removing any unnecessary complexity from the expression.

In the given exercise, after applying the quotient rule and chain rule, we ended up with an expression needing further simplification. Following these steps helps:

• Simplify the numerator by distributing and combining terms: \( 6x^2 - 3 - (6x^2 - 2x) = -3 + 2x \).
• Write the expression in a clear manner: \( \frac{-3 + 2x}{(2x^2 - 1)\sqrt{2x^2 - 1}} \).

Through simplification, the derivative becomes more manageable and can be easier to use in calculus problems, whether for further calculation or analysis.