Problem 22 - Thomas Calculus

Question

In Exercises \(19-22,\) find the values of the derivatives. $$ \left.\frac{d w}{d z}\right|_{z=4} \quad \text { if } \quad w=z+\sqrt{z} $$

Step-by-Step Solution

Answer

The derivative at \( z = 4 \) is \( \frac{5}{4} \).

Step 1: Identify the Function and Point

Given the function \( w = z + \sqrt{z} \), we need to find the value of the derivative \( \frac{dw}{dz} \) at \( z = 4 \).

Step 2: Apply Derivative Rules

Differentiate the function \( w = z + \sqrt{z} \) with respect to \( z \). - The derivative of \( z \) is \( 1 \).- The derivative of \( \sqrt{z} = z^{1/2} \) is \( \frac{1}{2}z^{-1/2} \) using the power rule.

Step 3: Calculate the Full Derivative

The derivative \( \frac{dw}{dz} \) is given by\[ \frac{dw}{dz} = \frac{d}{dz} (z) + \frac{d}{dz} \left(z^{1/2}\right) = 1 + \frac{1}{2}z^{-1/2}. \]

Step 4: Substitute the Value of z

Substitute \( z = 4 \) into the derivative function \[ \frac{dw}{dz} = 1 + \frac{1}{2} \times 4^{-1/2}. \]

Step 5: Simplify the Expression

Calculate the value of \( 4^{-1/2} \), which is \( \frac{1}{2} \), giving \[ \frac{dw}{dz} = 1 + \frac{1}{2} \times \frac{1}{2} = 1 + \frac{1}{4} = \frac{5}{4}. \]

Key Concepts

Derivative Rules

Calculus derivatives are a way to understand how functions change. In calculus, we often use derivative rules to find the rate at which something changes. These rules simplify the process of finding a derivative, making it easier to solve complex problems. There are several derivative rules to remember:

• The **Constant Rule** says that the derivative of a constant number is zero.
• The **Power Rule** helps us handle derivatives of powers of a variable, like \(x^n\).
• The **Sum Rule** lets us take the derivative of a sum by differentiating each term separately.
• The **Product Rule** and **Quotient Rule** are used when differentiating products and quotients.
Each rule caters to different types of functions and combining them can unlock solutions for almost any function you come across. Understanding when and how to apply these rules is the key to mastering derivatives.

Power Rule

The Power Rule is a fundamental derivative rule that makes differentiating powers of a variable straightforward. It states that if you have a function in the form \(f(x) = x^n\), the derivative \(f'(x)\) is \(nx^{n-1}\). This means you multiply by the power and then decrease the power by one.

Example of Using the Power Rule:

If you have \(z^{1/2}\), here's what happens:

• Identify the power: It's \(1/2\).
• Apply the Power Rule: Bring down the \(1/2\) and reduce the power by one, getting \(\frac{1}{2}z^{-1/2}\).

This simple rule helps us easily determine derivatives of polynomial and radical expressions. Getting comfortable with the Power Rule will greatly simplify your calculus journey.

Evaluating Derivatives

Once you've found the derivative function, evaluating derivatives involves finding the specific rate of change at a particular point. For example, if you're asked to find the derivative at \(z = 4\), you substitute this value into the derivative expression.

Steps to Evaluate:

• Find the derivative expression using the proper rules.
• Substitute the given value (like \(z = 4\)) into the derivative function.
• Simplify the expression to get the numerical value.

In our example, the derivative \(\frac{dw}{dz}\) at \(z = 4\) becomes \(1 + \frac{1}{2} \times 4^{-1/2}\), simplifying to \(\frac{5}{4}\). Being able to evaluate derivatives lets you understand how a function behaves at specific points.