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The power rule is a fundamental tool in calculus used to differentiate functions of the form \( x^n \), where \( n \) is any real number.
It's a simple yet powerful method that states: to find the derivative of \( x^n \), multiply the power by the coefficient and then subtract one from the power.
This can be expressed as: \[ \frac{d}{dx}x^n = nx^{n-1}\] Let's consider how the power rule applies to our example. After rewriting the function \( y = x^{3/2} + 4x^{1/2} + 3x^{-1/2} \), the power rule allows us to differentiate each term individually:

• For \( x^{3/2} \), the derivative is \( \frac{3}{2}x^{1/2} \).
• For \( 4x^{1/2} \), we have the derivative \( 2x^{-1/2} \).
• For \( 3x^{-1/2} \), we find \( -\frac{3}{2}x^{-3/2} \).

Using the power rule simplifies the process of differentiation especially when dealing with polynomial terms indicated by different exponents.

A derivative represents the rate at which a function is changing at any given point.
It's essentially the function's slope or steepness at specific points along its curve.
This is vital for understanding how functions behave, especially in the context of physics, engineering, and economics.
In essence, calculating a derivative helps gauge how a small change in \( x \) influences \( y \). In our example with the function \( y = \frac{x^2 + 4x + 3}{\sqrt{x}} \), finding \( y' \) means determining how the value of \( y \) changes with \( x \).
The process of differentiation involves using rules like the power rule, as applied in our solution:

• Step by step differentiation of terms \( x^{3/2}, 4x^{1/2}, \text{and } 3x^{-1/2} \) gives us insight about their respective rates of change.
• These individual derivatives are then combined into one expression to describe the overall rate of change for the entire function.

Derivatives are crucial for predicting behavior and understanding trends in various functions.

Simplifying expressions is an essential technique in calculus that makes the process of differentiation much more manageable.
The primary goal is to transform the function into a simpler form that is easier to work with.
In our exercise, we start with a relatively complex form: \( y = \frac{x^2 + 4x + 3}{\sqrt{x}} \). By simplifying each term separately:

• Convert \( \frac{x^2}{\sqrt{x}} \) into \( x^{3/2} \).
• Turn \( \frac{4x}{\sqrt{x}} \) into \( 4x^{1/2} \).
• Simplify \( \frac{3}{\sqrt{x}} \) as \( 3x^{-1/2} \).

This results in an expression, \( y = x^{3/2} + 4x^{1/2} + 3x^{-1/2} \), which is far more straightforward for applying differentiation techniques.
Simplifying expressions not only expedites differentiation but also reduces the likelihood of errors. By having a tidy form, using rules like the power rule becomes much simpler and more direct.