Problem 25 - Calculus for Biology and Medicine

Question

Use the formal definition to find the derivative of $$ y=\sqrt{x} $$

Step-by-Step Solution

Answer

The derivative of \( y = \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \).

Step 1: Recall the Derivative Definition

The formal definition of a derivative for a function \( f(x) \) is given by the limit \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \). We will use this to find the derivative of \( y = \sqrt{x} \).

Step 2: Identify \( f(x+h) \) and \( f(x) \)

For \( y = \sqrt{x} \), we have \( f(x) = \sqrt{x} \). Thus, \( f(x+h) = \sqrt{x+h} \). Now, substitute these into the derivative definition.

Step 3: Set Up the Derivative Expression

Substitute \( f(x+h) \) and \( f(x) \) into the limit expression: \( \frac{{f(x+h) - f(x)}}{h} = \frac{{\sqrt{x+h} - \sqrt{x}}}{h} \).

Step 4: Simplify the Expression Using Conjugates

To simplify, multiply the numerator and the denominator by the conjugate of the numerator: \( \frac{{\sqrt{x+h} - \sqrt{x}}}{h} \times \frac{{\sqrt{x+h} + \sqrt{x}}}{\sqrt{x+h} + \sqrt{x}} = \frac{{(\sqrt{x+h})^2 - (\sqrt{x})^2}}{h (\sqrt{x+h} + \sqrt{x})} = \frac{{(x + h) - x}}{h (\sqrt{x+h} + \sqrt{x})} \).

Step 5: Further Simplify and Take the Limit

Now simplify the expression: \( \frac{h}{h (\sqrt{x+h} + \sqrt{x})} = \frac{1}{\sqrt{x+h} + \sqrt{x}} \). Take the limit as \( h \to 0 \): \( \lim_{{h \to 0}} \frac{1}{\sqrt{x+h} + \sqrt{x}} = \frac{1}{2\sqrt{x}} \).

Step 6: Conclusion: Write the Derivative

Thus, the derivative of \( y = \sqrt{x} \) is \( \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \).

Key Concepts

Formal Definition of a Derivative

The formal definition of a derivative is a fundamental concept in calculus. It expresses how a function changes at any given point. For a function \( f(x) \), the derivative \( f'(x) \) is defined as the limit of the average rate of change of \( f(x) \) as the interval approaches zero. This is represented mathematically as:

• \( f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \)

This formula calculates the slope of the tangent line to the curve \( f(x) \) at any point \( x \). Understanding this provides insight into how quickly the function's values are changing at a particular moment. The derivative is indispensable in tracking rates of change across various disciplines like physics, economics, and beyond.

Limit Process

The limit process is a critical component of calculating derivatives. It helps us understand the behavior of functions as we approach a certain point. In the derivative's formal definition, we use the limit process to find \( f'(x) \) by examining the function's rate of change as \( h \) approaches zero.
When finding the derivative of \( y = \sqrt{x} \), we plug \( f(x+h) = \sqrt{x+h} \) and \( f(x) = \sqrt{x} \) into the limit expression:

• \( \lim_{{h \to 0}} \frac{{\sqrt{x+h} - \sqrt{x}}}{h} \)

This reveals the instantaneous rate of change of \( \sqrt{x} \). The limit process ensures that the derivative provides an exact rather than approximate rate of change.

Differentiation Techniques

One of the essential techniques in differentiation involves manipulating expressions to apply the derivative definition successfully. In our case of finding the derivative of \( y = \sqrt{x} \), we need to simplify \( \frac{{\sqrt{x+h} - \sqrt{x}}}{h} \) using a clever method known as rationalizing with conjugates.
First, multiply the expression by the conjugate of the numerator:

• \( \frac{{\sqrt{x+h} - \sqrt{x}}}{h} \times \frac{{\sqrt{x+h} + \sqrt{x}}}{\sqrt{x+h} + \sqrt{x}} \)

This results in removing the radicals from the numerator, turning the problem into applying polynomial division within the limits:

• \( \frac{{h}}{{h(\sqrt{x+h} + \sqrt{x})}} \)

Understanding such techniques is vital, as they frequently appear when dealing with more complex functions.

Simplifying Expressions

Simplifying expressions is a significant step in applying the derivative definition. It involves breaking down complex algebraic expressions into simpler components. During differentiation, simplifying expressions often helps in handling limits more straightforwardly.
In finding the derivative of \( y = \sqrt{x} \), simplifying through multiplying by the conjugate leads to:

• \( \frac{h}{h(\sqrt{x+h} + \sqrt{x})} = \frac{1}{\sqrt{x+h} + \sqrt{x}} \)

As \( h \to 0 \), the expression resolves to its simplest form:

• \( \frac{1}{2\sqrt{x}} \)

Simplifying the expression allowed us to compute the derivative accurately, demonstrating the need for such skills in calculus. They help to work through the algebraic intricacies that often arise in differentiation.