Problem 77 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=\frac{a x^{2}}{4+x^{2}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{8ax}{(4+x^2)^2} \).

Step 1: Apply the Quotient Rule

To differentiate the given function \( f(x) = \frac{ax^2}{4 + x^2} \), we use the quotient rule for derivatives, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \].For our function, identify \( u(x) = ax^2 \) and \( v(x) = 4 + x^2 \).

Step 2: Differentiate the Numerator

Differentiate \( u(x) = ax^2 \) with respect to \( x \). The derivative is:\[ u'(x) = 2ax \].

Step 3: Differentiate the Denominator

Differentiate \( v(x) = 4 + x^2 \) with respect to \( x \). The derivative is:\[ v'(x) = 2x \].

Step 4: Plug into the Quotient Rule Formula

Substitute the derivatives found in the previous steps into the quotient rule formula:\[ f'(x) = \frac{(2ax)(4 + x^2) - (ax^2)(2x)}{(4 + x^2)^2} \].Now simplify the expression.

Step 5: Simplify the Expression

Simplify the numerator:\[(2ax)(4 + x^2) = 8ax + 2ax^3 \]\[(ax^2)(2x) = 2ax^3 \]Subtract these to get the numerator:\[ 8ax + 2ax^3 - 2ax^3 = 8ax \]So, the simplified derivative is:\[ f'(x) = \frac{8ax}{(4 + x^2)^2} \].

Key Concepts

Understanding the Quotient Rule

The quotient rule is a method used in calculus for finding the derivative of a function that is the ratio of two differentiable functions. When you have a function of the form \( \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable functions of \( x \), the derivative of this function with respect to \( x \) is given by:\[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]In the provided example, \( f(x) = \frac{ax^2}{4 + x^2} \), this rule is applied to differentiate \( f(x) \). The individual parts are:

• \( u(x) = ax^2 \) with its derivative \( u'(x) = 2ax \).
• \( v(x) = 4 + x^2 \) with its derivative \( v'(x) = 2x \).

The quotient rule helps in systematically applying these derivatives to find \( f'(x) \). Remember, it's important to precisely follow the rule to ensure all terms are correctly differentiated and placed in their respective positions in the formula.

Derivative Calculation

Calculating derivatives is the core process in determining how a function changes with respect to another variable, typically \( x \). The derivative represents the slope of the tangent line to the curve of the function at any point. For the function \( f(x) = \frac{ax^2}{4 + x^2} \), we need to find derivatives of both the numerator and the denominator separately before applying them in the quotient rule.When differentiating the numerator \( u(x) = ax^2 \), the power rule is useful. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). So:\[ u'(x) = 2ax \]For the denominator \( v(x) = 4 + x^2 \), applying the power rule again,\[ v'(x) = 2x \]These individual steps are fundamental in obtaining the overall derivative through the quotient rule.

Simplifying Expressions

After applying the quotient rule and calculating the necessary derivatives, the next crucial step is simplifying the resulting expression. Simplification helps make the expression easier to interpret and use for further calculations. For our function, the expression obtained after applying the quotient rule is:\[ f'(x) = \frac{(2ax)(4 + x^2) - (ax^2)(2x)}{(4 + x^2)^2} \]To simplify, first expand the terms in the numerator:

• \((2ax)(4 + x^2) = 8ax + 2ax^3\)
• \((ax^2)(2x) = 2ax^3\)

By subtracting, we get:\[ 8ax + 2ax^3 - 2ax^3 = 8ax \]This leads to the simplified derivative:\[ f'(x) = \frac{8ax}{(4 + x^2)^2} \]Simplifying expressions involves combining like terms and ensuring that the expression is as concise as possible. This step is essential in mathematical problem-solving, as it provides the most straightforward representation of the solution.