Problem 16 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=8 \sqrt{x} $$

Step-by-Step Solution

Answer

The derivative of \( y = 8 \sqrt{x} \) is \( \frac{4}{\sqrt{x}} \).

Step 1: Identify the form of the expression

The function given is \( y = 8 \sqrt{x} \). Recognize that \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).

Step 2: Rewrite the function using exponents

Express \( y = 8 \sqrt{x} \) using exponents: \( y = 8x^{1/2} \).

Step 3: Apply the power rule for differentiation

The power rule states \( \frac{d}{dx} x^n = nx^{n-1} \). Differentiate \( y = 8x^{1/2} \) using this rule: \( \frac{d}{dx} (8x^{1/2}) = 8 \times \frac{1}{2} x^{1/2-1} \).

Step 4: Simplify the derivative

Simplify the expression: \( \frac{d y}{d x} = 4 x^{-1/2} \).

Step 5: Rewrite the expression in radical form

Convert \( 4 x^{-1/2} \) back to radical form: \( \frac{4}{\sqrt{x}} \).

Key Concepts

Power Rule

The power rule is an essential tool in calculus for differentiating functions of the form \( x^n \), where \( n \) is any real number. It simplifies the process of finding the derivative of functions that can be expressed in this manner. According to the power rule, the derivative of \( x^n \) is given by \( nx^{n-1} \).

For instance, if you have a function like \( y = 8x^{1/2} \), the power rule can be applied to differentiate it. First, you multiply the exponent \( \frac{1}{2} \) by the coefficient \( 8 \), then you subtract one from the exponent. So the derivative becomes \( \frac{d y}{d x} = 8 \times \frac{1}{2} x^{1/2-1} \). This results in \( 4x^{-1/2} \).

Remember to always adjust the exponent by subtracting one after multiplying by the original exponent. This simple yet powerful rule makes it quick to differentiate a wide variety of polynomial expressions.

Differentiation

Differentiation is the mathematical process of finding the derivative of a function. It measures how a function's output value changes concerning its input value. In simple terms, differentiation tells you the rate at which something changes. This is extremely useful in understanding trends and dynamic systems in fields such as physics, engineering, and economics.

When differentiating a function like \( y = 8 \sqrt{x} \), you first rewrite the square root in exponent form as \( y = 8x^{1/2} \). This makes it easier to apply differentiation rules such as the power rule. Differentiating gives you \( \frac{d y}{d x} = 4x^{-1/2} \), which tells you how the function \( y \) changes as \( x \) changes.

In practical applications, differentiation can be used to find slopes of curves, optimize functions (such as finding maximum or minimum values), and solve practical problems related to rates of change.

Exponent Rules

Exponent rules are basic algebraic guidelines that help us perform operations involving exponents systematically. Understanding these rules makes it easier to manipulate and simplify expressions involving powers.

Some key exponent rules include:

• Product of Powers Rule: \(x^a \cdot x^b = x^{a+b}\)
• Quotient of Powers Rule: \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of a Power Rule: \((x^a)^b = x^{ab}\)

In our context, consider the function originally given as \( y = 8 \sqrt{x} \). Recognizing that \( \sqrt{x} \) is equivalent to \( x^{1/2} \) demonstrates the use of exponents to simplify expressions before applying calculus techniques like differentiation. In the solution, the exponent rule was used to help transform \( 8 \sqrt{x} \) into a form that could be easily differentiated using the power rule.

Thus, mastery of exponent rules is not only crucial for simplifying expressions but also for setting up problems in a way that makes them easier to solve.