Problem 37 - Calculus - AP Edition

Question

Find \(d y / d x\) $$y=\left(\frac{x-5}{2 x+1}\right)^{3}$$

Step-by-Step Solution

Answer

\(\frac{33(x-5)^2}{(2x+1)^4}\) is the derivative, \(d y/d x\).

Step 1: Identify the Differentiation Technique

The function \(y = \left(\frac{x-5}{2x+1}\right)^3\) is a composition of functions, which means we need to use the chain rule for differentiation. The outer function here is \(u^3\) and the inner function is \(u = \frac{x-5}{2x+1}\).

Step 2: Differentiate the Outer Function

Differentiate the outer function \(u^3\) with respect to \(u\) using the power rule. The derivative is \(3u^2\).

Step 3: Differentiate the Inner Function

The inner function is a quotient: \(u = \frac{x-5}{2x+1}\). Use the quotient rule to differentiate it:\(\frac{d}{dx}\left(\frac{x-5}{2x+1}\right) = \frac{(2x+1) \cdot 1 - (x-5) \cdot 2}{(2x+1)^2} = \frac{2x+1 - 2x + 10}{(2x+1)^2} = \frac{11}{(2x+1)^2}\).

Step 4: Apply the Chain Rule

Combine the results from Steps 2 and 3 using the chain rule. The differential \(\frac{dy}{dx}\) is obtained by multiplying the derivative of the outer function by the derivative of the inner function:\(\frac{dy}{dx} = 3 \left(\frac{x-5}{2x+1}\right)^2 \cdot \frac{11}{(2x+1)^2} = \frac{33(x-5)^2}{(2x+1)^4}\).

Step 5: Simplify the Result

The expression \(\frac{33(x-5)^2}{(2x+1)^4}\) is already in a simplified form, which is the final derivative.

Key Concepts

Quotient Rule

The quotient rule is an essential differentiation technique used when dealing with functions in the form of a fraction, where one function is divided by another. It helps us find the derivative of the ratio of two differentiable functions. For any two functions \(u(x)\) and \(v(x)\), the derivative of their quotient \(\frac{u}{v}\) is given by:\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\]Here:

• \(u'\) is the derivative of the numerator \(u\).
• \(v'\) is the derivative of the denominator \(v\).
• \(v^2\) is the square of the denominator \(v\).

In our exercise, we used the quotient rule to find the derivative of \(\frac{x-5}{2x+1}\), where \(u = x-5\) and \(v = 2x+1\). After applying the quotient rule, it becomes clear how we arrived at the expression \(\frac{11}{(2x+1)^2}\) in Step 3 of the solution.

Power Rule

The power rule is a fundamental tool in differentiation that simplifies finding the derivative of a function with a variable raised to a power. If you have a function in the form \(u^n\), the power rule states that:\[\frac{d}{dx} (u^n) = n \cdot u^{n-1} \cdot u'\]This rule is particularly useful in our exercise when we differentiate the outer function \(u^3\). By applying the power rule, we discovered the derivative to be \(3u^2\). Remember, the power rule simplifies calculating derivatives where powers are involved by directly giving you the multiplier (the exponent) in front multiplied by \(u\), with its power reduced by one.
The power of applying this rule effectively allows us to handle polynomials and expressions with variable exponents swiftly.

Differentiation Techniques

Differentiation techniques encompass various methods used to find the derivative of a function. Each function type, whether it be a simple polynomial, a complex fraction, or even trigonometric, requires a technique tailored to its structure.

• Chain Rule: Used when dealing with composite functions, as seen in our solution. The chain rule connects the derivative of the outer function with that of the inner function, ensuring a seamless integration of both derivatives.
• Quotient Rule: Specifically used for differentiating functions presented as ratios. It calculates derivatives by balancing the influence of both numerator and denominator.
• Power Rule: Simplifies the process by directly applying the exponent's role in differentiation, vital for polynomials.

These techniques are the building blocks of calculus, creating the foundation needed to tackle complex differentiation problems. In our example, each technique plays a critical role in completely understanding how to derive the function \(y = \left(\frac{x-5}{2x+1}\right)^3\). By mastering these differentiation techniques, one becomes adept at handling diverse calculus challenges.