Problem 11 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ f(y)=\left(\frac{y-7}{y+2}\right)^{2} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{18(y-7)}{(y+2)^3} \)

Step 1: Identify the Outer Function

First, identify the outer function and inner function. The given function is \[ f(y) = \bigg(\frac{y-7}{y+2}\bigg)^2 \]Here, the outer function is \[ u^2 \text{where }\text{u}=\frac{y-7}{y+2}\]

Step 2: Differentiate the Outer Function

Differentiate the outer function with respect to the inner function. The derivative of \[ u^2 \text{with respect to}\text{u}\text{ is }2u \]So,\[ f'(u) = 2 \bigg(\frac{y-7}{y+2}\bigg) \]

Step 3: Differentiate the Inner Function

Now differentiate the inner function \(\frac{y-7}{y+2}\) with respect to \(y\). Use the quotient rule for this: \[\bigg(\frac{u}{v}\bigg)' = \frac{v\frac{d}{dy}u - u\frac{d}{dy}v}{v^2}\]Here, \(u = y-7\) and \(v = y+2\). So,\[\frac{d}{dy}(y-7) = 1\] \[\frac{d}{dy}(y+2) = 1\]Substitute these into the quotient rule:\[\bigg(\frac{y-7}{y+2}\bigg)' = \frac{(y+2) \cdot 1 - (y-7) \cdot 1}{(y+2)^2} = \frac{y+2-y+7}{(y+2)^2} = \frac{9}{(y+2)^2}\]

Step 4: Apply the Chain Rule

Now, use the chain rule to find the derivative of the composite function. The chain rule states that \[ f'(y) = f'(u) \cdot u'(y) \]Here \(f'(u) = 2 \bigg(\frac{y-7}{y+2}\bigg)\) and \(u'(y) = \frac{9}{(y+2)^2}\). So,\[ f'(y) = 2 \bigg(\frac{y-7}{y+2}\bigg) \cdot \frac{9}{(y+2)^2} = \frac{18(y-7)}{(y+2)^3} \]

Key Concepts

quotient rule

When dealing with the derivative of a function that is the division of two other functions, the quotient rule is essential. For the functions, let's call the numerator \(u\) and the denominator \(v\). To find the derivative of this quotient, use the formula: \[ \left(\frac{u}{v}\right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \] This formula tells us to:

• Differentiate the numerator \(u\) and the denominator \(v\)
• Multiply the derivative of the numerator by the original denominator
• Multiply the derivative of the denominator by the original numerator
• Subtract these two products
• Divide everything by the square of the denominator

Applying this to our exercise, with \(u = y-7\) and \(v = y+2\), simplifies the original problem. Remember: practicing more examples will solidify this important rule!

chain rule

The chain rule is a technique for finding the derivative of a composite function. When you have one function inside another, this is where the chain rule shines. Imagine we have a composite function \(f(g(x))\). The chain rule states that:

• First, differentiate the outer function \(f\) as if the inner function \(g(x)\) were just a variable.
• Then, multiply by the derivative of the inner function \(g(x)\).

The formula is \[ \frac{d}{dx}f(g(x)) = f'(g(x)) \frac{dg}{dx} \] In our exercise, the outer function is \(u^2\) and the inner function is \( \frac{y-7}{y+2} \). By applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. This systematic approach simplifies complex functions.

composite function

A composite function is when you have one function inside another, like \(f(g(x))\). In our example, \(f(y) = \left( \frac{y-7}{y+2} \right)^2\), the outer function is a square, and the inner function is a fraction! To differentiate such a function, we first identify the inner and outer functions.
After recognizing these layers, use the chain rule to take one derivative at a time, starting from the outermost layer and working inward. Recognizing the composition helps break down the problem into simpler parts. Practice is key! Learning to identify the composite layers quickly will greatly improve problem-solving efficiency. Each step peels back a layer, making complex derivatives more manageable!