Problem 4 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=x^{-12}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = -12 \cdot x^{-13} \).

Step 1: Identify the Power Rule for Derivatives

The power rule states that if you have a function of the form \( y = x^n \), then its derivative is found using \( \frac{dy}{dx} = n \cdot x^{n-1} \). This rule can be applied directly to our function \( y = x^{-12} \).

Step 2: Differentiate Using the Power Rule

Apply the power rule where \( n = -12 \): \[ \frac{dy}{dx} = -12 \cdot x^{-12-1} \].

Step 3: Simplify the Expression

The expression from Step 2 simplifies to \[ \frac{dy}{dx} = -12 \cdot x^{-13} \]. This is the derivative of the function \( y = x^{-12} \).

Key Concepts

Understanding Derivatives

When tackling the concept of a derivative, think of it as a way to measure how one quantity changes as another one affecting it changes. It's all about rates of change. In a more technical sense, the derivative of a function describes the function's rate of change at any given point on its curve. This is especially handy in calculus for finding things like velocity, acceleration, and more. Let's dive into a few key points of understanding derivatives:

• Derivatives can symbolize slopes of tangent lines to curves.
• They tell us how a small change in one variable affects another.
• By taking the derivative of a function, you often get another function that predicts immediate changes.

The derivative is a foundational tool in calculus, enabling us to explore the changing world mathematically! Remember, not all functions will have derivatives everywhere, but for those that do, derivatives offer a powerful method of analysis.

Learning Differentiation

Differentiation is the process through which we find derivatives. It's akin to a mathematical detective work where we identify rates of change and slopes of curves we care about. Differentiation can be straightforward when using rules like the power rule, but it may also require applying multiple rules or techniques for more complex functions.
Some key ideas in differentiation include:

• The process of finding a derivative involves calculations around limits.
• Differentiation rules, such as the sum rule and product rule, help process derivatives neatly.
• Techniques like implicit differentiation and using substitution aid in tackling complex functions.

In the problem at hand, differentiation was directly applied using the power rule, showcasing its efficiency for straightforward functions. Always remember, learning differentiation fundamentally changes how one approaches dynamic mathematical models.

Mastering Exponent Rules

Mastering exponent rules is crucial for applying the power rule effectively. These rules simplify expressions with exponents and assist in differentiation processes. Essentially, exponent rules allow for simplification of expressions where variables have powers.Here are a few essential exponent rules:

• Product of Powers Rule: When multiplying like bases, add the exponents. \(x^a \cdot x^b = x^{a+b}\)
• Power of a Power Rule: When raising a power to another power, multiply the exponents. \((x^a)^b = x^{a \cdot b}\)
• Negative Exponent Rule: Negative exponents indicate reciprocals. \(x^{-a} = \frac{1}{x^a}\)

In the given exercise, the negative exponent rule helped streamline the differentiation. The power rule works hand in hand with these exponent rules, ensuring clear and accurate results when deriving functions.