Question

Find \(D_{x} y\). $$ y=\frac{(x+1)^{2}}{3 x-4} $$

Short answer

The derivative is \( D_{x} y = \frac{3x^2 - 8x - 11}{(3x - 4)^2} \).

Step-by-step solution

Identify the Derivative Rule

The function is a quotient of two functions, so we will use the quotient rule to find the derivative. The quotient rule is: if \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^{2}} \).

Define Functions and Derivatives

Identify the numerator and denominator functions as \( u = (x+1)^2 \) and \( v = 3x - 4 \). Next, find the derivatives: \( \frac{du}{dx} = 2(x+1) \) and \( \frac{dv}{dx} = 3 \).

Apply the Quotient Rule

Substitute \( u, \frac{du}{dx}, v, \) and \( \frac{dv}{dx} \) into the quotient rule: \[ D_{x} y = \frac{(3x - 4) \cdot 2(x+1) - (x+1)^2 \cdot 3}{(3x - 4)^2}. \]

Simplify the Derivative Expression

Begin simplifying the expression from the numerator:\[ D_{x} y = \frac{[6x(x+1) + 2(x+1)(-4)] - 3(x+1)^2}{(3x - 4)^2}. \]This becomes:\[ D_{x} y = \frac{(6x^2 + 6x - 8x - 8) - (3x^2 + 6x + 3)}{(3x - 4)^2}. \]

Combine Like Terms

Combine the like terms in the simplified expression:\[ D_{x} y = \frac{6x^2 - 2x - 8 - 3x^2 - 6x - 3}{(3x - 4)^2}. \]This results in:\[ D_{x} y = \frac{3x^2 - 8x - 11}{(3x - 4)^2}. \]

Key concepts

Derivative

The derivative of a function gives us the rate at which its value changes in response to changes in the input. In simple terms, think of the derivative as a tool that tells us how steep a graph is at any point. This can represent how fast something is speeding up or slowing down.
In our case, we need to find the derivative of a function that is given as a quotient.
To do this effectively, one must first identify which part of the function corresponds to the numerator and which to the denominator.
This recognition is crucial for applying the right strategies to compute derivatives correctly.

Simplification

Simplification is crucial after computing the initial derivative, especially when dealing with complex expressions. Simplifying means making the equation or expression easier and more straightforward without changing its meaning.
It involves combining like terms and reducing fractions if possible.
This step is about organizing the equation to make it easier to interpret and solve further.

• Helps in error-checking: Simplified expressions are usually less prone to calculation errors.
• Makes further calculations and integrations easier.

Here, after applying the quotient rule, simplifying our derivative improves clarity and eases further usage.

Quotient of Functions

The quotient of functions involves taking one function and dividing it by another. In calculus, this is common, and understanding how to handle these operations is key.
The basic formula for differentiating a quotient function is given by the quotient rule:\[ \frac{dy}{dx} = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]where \(u\) is the numerator and \(v\) is the denominator.
It's essential because many real-world applications involve rates of changes where one quantity affects another, creating a natural division situation, or `quotient`.

• The correct application of this rule ensures accuracy in finding derivatives.
• Remember always to square the denominator when applying the quotient rule.

Being familiar with this concept is important in navigating through different calculus problems effectively.