Question

Find \(D_{x} y\). $$ y=\frac{2 x-3}{\left(x^{2}+4\right)^{2}} $$

Short answer

Use the quotient rule and simplify the expression to find the derivative, \( D_x y \).

Step-by-step solution

Identify the Differentiation Technique

The function \( y = \frac{2x - 3}{(x^2 + 4)^2} \) is a rational function, and we will use the quotient rule to differentiate it. The quotient rule states that for \( y = \frac{u}{v} \), the derivative \( D_x y = \frac{v \cdot D_x u - u \cdot D_x v}{v^2} \). Here, \( u = 2x - 3 \) and \( v = (x^2 + 4)^2 \).

Differentiate the Numerator

Differentiate \( u = 2x - 3 \) with respect to \( x \). The derivative of \( u \) is \( D_x u = 2 \).

Differentiate the Denominator

The function \( v = (x^2 + 4)^2 \) is a composite function, so we'll use the chain rule. Let \( z = x^2 + 4 \), then \( v = z^2 \). The derivative of \( v \) is \( D_x v = 2z \cdot D_x z \). Now, \( D_x z = 2x \), so \( D_x v = 2(x^2 + 4) \cdot 2x = 4x(x^2 + 4) \).

Apply the Quotient Rule

Using the quotient rule: \(D_x y = \frac{(x^2 + 4)^2 \cdot 2 - (2x - 3) \cdot 4x(x^2 + 4)}{((x^2 + 4)^2)^2}\)Simplify the expression by expanding the terms in the numerator.

Simplify the Derivative Expression

Expand and simplify the expression:The first term: \((x^2+4)^2 \cdot 2 = 2(x^4 + 8x^2 + 16)\)The second term: \((2x - 3) \cdot 4x(x^2 + 4) = (2x - 3)(4x^3 + 16x)\)Now, plug these back into the expression and simplify it to get the final derivative.

Key concepts

Quotient rule

The quotient rule is a technique you use when differentiating a function that can be expressed as one function divided by another. It is very handy when dealing with ratios of functions, especially rational functions. The basic idea is to take the derivative of each part, the numerator and the denominator separately, and then combine them using a specific formula.

The quotient rule formula is:

• If you have a function \( y = \frac{u}{v} \), where both \( u \) and \( v \) are differentiable, the derivative \( D_x y \) is given by \( D_x y = \frac{v \cdot D_x u - u \cdot D_x v}{v^2} \).
• In other words, multiply the derivative of the numerator by the denominator, subtract the product of the numerator and the derivative of the denominator, and divide everything by the square of the denominator.

Use the quotient rule whenever you see a function that looks like a fraction of two differentiable functions. Always remember to differentiate both the top and bottom parts of the fraction separately before applying the rule. This helps ensure accuracy and avoid mistakes.

Chain rule

The chain rule is a method used to differentiate composite functions. A composite function is one function inside of another, essentially a function within a function. When you differentiate using the chain rule, you're effectively peeling back the layers of this composite to find the derivative.

To use the chain rule:

• Imagine you have a function \( v = (x^2 + 4)^2 \). Here, the outer function is \( z^2 \) and the inner function is \( x^2 + 4 \).
• Start by differentiating the outer function, treating the inner function as a single entity. This gives you \( 2z \).
• Next, differentiate the inner function. For \( z = x^2 + 4 \), this derivative is \( 2x \).
• Finally, apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: \( D_x v = 2(x^2 + 4) \cdot 2x = 4x(x^2 + 4) \).

This method is powerful because it allows for precise differentiation of functions that seem complex at first glance, by breaking them down into more manageable parts.

Rational function differentiation

Rational functions are fractions where both the numerator and the denominator are polynomials. Differentiating rational functions typically involves the quotient rule, because they consist of one polynomial divided by another. Understanding the behavior of rational functions is crucial in calculus, as they appear frequently across different problems.

To effectively differentiate a rational function:

• Recognize the function is in the form \( \frac{u}{v} \), where \( u \) is the numerator and \( v \) is the denominator. Both \( u \) and \( v \) contain terms that need to be differentiated.
• Apply the quotient rule to find the derivative efficiently. This involves using both the derivatives you've found from the numerator and the denominator.
• In some cases, the functions within the numerator or the denominator might require chain rule application, especially if they are composite functions themselves.
• Simplify the resulting derivative expression by expanding and combining like terms for clarity and accuracy.

Understanding these steps provides a firm foundation for tackling rational functions, enabling you to handle more intricate calculus problems with ease.