Problem 11 - Brief Applied Calculus

Question

Find the derivative of each function. $$ h(x)=\frac{3}{x^{2}} $$

Step-by-Step Solution

Answer

The derivative of \( h(x) = \frac{3}{x^2} \) is \( h'(x) = -6x^{-3} \).

Step 1: Rewrite the Function

Start by rewriting the function to make differentiating easier. The function is given as \( h(x) = \frac{3}{x^2} \). We can rewrite this as \( h(x) = 3x^{-2} \). This allows us to use the power rule more straightforwardly.

Step 2: Identify the Power Rule

The power rule for differentiation states that if \( f(x) = ax^n \), then the derivative \( f'(x) = anx^{n-1} \). We will apply this rule to our rewritten function.

Step 3: Differentiate Using the Power Rule

Apply the power rule to \( h(x) = 3x^{-2} \). According to the power rule, the derivative is \( h'(x) = 3(-2)x^{-2-1} = -6x^{-3} \).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding derivatives, especially when dealing with polynomial terms. It's a straightforward method to determine the derivative of a function of the form \( f(x) = ax^n \). Here's how it works:

• If you have a term \( ax^n \), its derivative, using the power rule, is \( anx^{n-1} \).
• The rule is applied by multiplying the exponent \( n \) by the coefficient \( a \), then reducing the exponent by one.
• For example, if you had \( 3x^{-2} \), applying the power rule gives \( -6x^{-3} \).

Understanding the power rule is crucial because it simplifies the process of differentiation, especially for students new to calculus. The simplicity of just two steps in the power rule—multiply and reduce—makes it one of the most accessible differentiation techniques.

Differentiation

Differentiation is the process in calculus used to find the derivative of a function. The derivative represents the rate at which a function is changing at any given point and is a central concept in calculus. Here’s why differentiation is so important:

• It helps us understand how quantities change over time or space. For instance, velocity is a derivative of the position with respect to time.
• In simple terms, differentiation provides the slope of a function at any given point, enlightening us on how steep the graph of the function is at that point.
• Applying differentiation to a function like \( h(x) = 3x^{-2} \) efficiently provides insights into its behaviour and allows for practical applications, such as graphing.

Mastering differentiation involves understanding various rules including the power rule, the product rule, and more. These rules make it possible to handle increasingly complex functions.

Functions

Functions are one of the building blocks of calculus, embodying relationships between sets of numbers. In calculus, a function expresses how one variable depends on another. Here’s a fundamental look at functions:

• A function is typically written as \( f(x) \), indicating that \( f \) is a function of \( x \).
• This notation means that for each value of \( x \), the function gives a unique output \( f(x) \).
• For example, the function \( h(x) = \frac{3}{x^2} \) defines a specific mathematical rule that relates \( x \) to \( h(x) \).

Understanding functions is essential as they form the basis for tasks like differentiation. When working with functions, especially those that aren't simple polynomials, rewriting them (like converting a fraction to a negative power) often simplifies the differentiation process. Thus, grasping the concept of functions and their behaviours is vital for success in mathematics.