Problem 24 - Applied Calculus

Question

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ w=\frac{3 z}{1+2 z} $$

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{3}{(1 + 2z)^2} \).

Step 1: Identify the Formula for Derivative of a Quotient

To find the derivative of a function that is a quotient of two functions, we use the quotient rule. For a function of the form \( \frac{u}{v} \), the derivative \( \frac{d}{dz}\left(\frac{u}{v}\right) \) is given by:\[\frac{v \cdot \frac{du}{dz} - u \cdot \frac{dv}{dz}}{v^2}\]where \( u = 3z \) and \( v = 1 + 2z \) in our problem.

Step 2: Calculate Derivatives of the Numerator and Denominator

Next, find the derivatives \( \frac{du}{dz} \) and \( \frac{dv}{dz} \). - For \( u = 3z \), \( \frac{du}{dz} = 3 \).- For \( v = 1 + 2z \), \( \frac{dv}{dz} = 2 \).

Step 3: Apply the Quotient Rule Formula

Substitute \( u \), \( v \), \( \frac{du}{dz} \), and \( \frac{dv}{dz} \) into the quotient rule formula:\[\frac{(1 + 2z) \cdot 3 - 3z \cdot 2}{(1 + 2z)^2}\]Simplify the expression in the numerator.

Step 4: Simplify the Numerator

Perform the arithmetic in the numerator:\[(1 + 2z) \cdot 3 - 3z \cdot 2 = 3 + 6z - 6z = 3\]So, the simplified expression is \( \frac{3}{(1 + 2z)^2} \).

Step 5: Write the Final Derivative

The derivative of \( w = \frac{3z}{1+2z} \) with respect to \( z \) is:\[\frac{dw}{dz} = \frac{3}{(1 + 2z)^2}\]This is the final simplified result.

Key Concepts

Quotient Rule

When you're working with derivatives in calculus and you need to find the derivative of a quotient, the quotient rule becomes your best friend. Imagine you have a function that is a fraction, like \( \frac{u}{v} \), where both \( u \) and \( v \) are themselves functions of \( z \). The quotient rule gives us a systematic way to find the derivative of such a function.Here's how the quotient rule works:

• You first calculate the derivative of the top function (numerator), \( u \).
• Next, you find the derivative of the bottom function (denominator), \( v \).
• Then, plug these into the formula: \[ \frac{d}{dz}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dz} - u \cdot \frac{dv}{dz}}{v^2} \]

This formula tells us to multiply the derivative of \( u \) by \( v \) and subtract the product of \( u \) and the derivative of \( v \). You then divide by \( v^2 \). It's like a take-turns process for the numerator and denominator. Just remember that the order and sign in the formula are crucial! Doing these steps ensures you've handled both the numerator and the denominator properly when they're working together in a fraction.

Derivative Calculation

Derivative calculation sounds intimidating, but it's truly just applying a set of rules repeatedly. To keep things simple, let's focus on the components we need to handle a quotient.In our example, \( u = 3z \) and \( v = 1 + 2z \). The derivatives are calculated as follows:

• The derivative of \( u = 3z \) with respect to \( z \) is straightforward: \( \frac{du}{dz} = 3 \).
• Similarly, the derivative of \( v = 1 + 2z \) with respect to \( z \) is \( \frac{dv}{dz} = 2 \).

These simple derivatives come from basic differentiation rules:

• For \( az \), where \( a \) is a constant, the derivative is just \( a \).
• For a constant, the derivative is zero.

Once we have these derivative components, we can move to apply the quotient rule effectively, plugging these values into the formula to get our derivative expression. The key is to move methodically and apply each rule correctly.

Simplifying Expressions

After applying the quotient rule in calculus, you often end up with an expression that needs simplifying. Simplifying not only makes the expression look neat, but also makes further calculations easier.In our scenario, after applying the quotient rule, we get an expression:\[ \frac{(1 + 2z) \cdot 3 - 3z \cdot 2}{(1 + 2z)^2} \]Let's break it down:

• The numerator simplifies when you distribute and combine terms: \((1 + 2z) \cdot 3 = 3 + 6z\) and \(3z \cdot 2 = 6z\).
• You then subtract the results: \(3 + 6z - 6z = 3\).

The denominators stay as they are: \((1 + 2z)^2\).Our final simplified derivative becomes \( \frac{3}{(1 + 2z)^2} \). Simplifying expressions is about meticulously handling arithmetic and algebra so the derivative expression is as tidy and accessible as possible. It makes a bigger difference than you might think, especially in more complicated problems!