Problem 15 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=4 \sqrt{x} $$

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = \frac{2}{\sqrt{x}} \)

Step 1: Understand the Function

We are given the function \( y = 4 \sqrt{x} \). This function involves a square root of \( x \), which can be rewritten using exponent notation to simplify differentiation.

Step 2: Rewrite the Function

Rewrite the square root as a power: \( y = 4 \cdot x^{1/2} \). This step helps in applying the derivative rule for functions of the type \( x^n \).

Step 3: Apply the Power Rule

To find the derivative of \( y = 4x^{1/2} \), use the power rule: \( \frac{d}{dx} [x^n] = n \cdot x^{n-1} \).

Step 4: Differentiate the Function

Apply the power rule to differentiate: \( \frac{d}{dx} [4x^{1/2}] = 4 \cdot \frac{1}{2} \cdot x^{-1/2} \).

Step 5: Simplify the Derivative

Simplify the expression: \( \frac{d y}{d x} = 2x^{-1/2} \). Rewriting with a square root gives \( \frac{d y}{d x} = \frac{2}{\sqrt{x}} \).

Key Concepts

Power Rule

The Power Rule is a fundamental aspect of differentiation that simplifies finding derivatives of functions with exponents. Whenever you have a power of a variable, say - \( x^n \), the Power Rule states that the derivative is obtained by:- multiplying by the power (\( n \)),- and then reducing the power by one.
This is mathematically represented as:- \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \).
For example, when differentiating \( x^3 \), you multiply by the exponent, 3, and decrease the exponent by 1, resulting in \( 3x^2 \).
This rule is crucial for quickly handling derivatives of polynomials and any expressions that can be rewritten with exponents. It's simple yet powerful, making differentiation more efficient.

Square Root Function

The square root function is a specific type of mathematical function that can be quite common in calculus. It is generally expressed as:- \( \sqrt{x} \).
For differentiation purposes, it is beneficial to transform it into exponent notation. This is because operations on powers are easier.
The square root of a number or variable corresponds to raising that number or variable to the power of \( \frac{1}{2} \). Therefore:- \( \sqrt{x} = x^{1/2} \).
This transformation allows us to apply the Power Rule, making differentiation a smooth process. Whenever you encounter a square root in calculus, consider using this conversion to streamline finding derivatives.

Exponent Notation

Exponent Notation is a way to express numbers, particularly when dealing with roots and powers, in a more uniform way. It involves writing expressions in the form:- \( x^n \), where \( n \) is any real number.
Some key points about exponent notation include:

• Converting roots to powers. For example, \( \sqrt{x} \) is \( x^{1/2} \).
• Remembering laws of exponents, such as multiplying powers (\( x^a \cdot x^b = x^{a+b} \)).
• Knowing how to handle negative exponents, like \( x^{-n} = \frac{1}{x^n} \).

Using exponent notation effectively can make operations like differentiation more straightforward, particularly when combined with the power rule. It allows handling of both simple and complex functions with ease during calculus operations.