Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\) using the rules of this section. $$ y=-3 x^{-4} $$

Step-by-Step Solution

Answer

The derivative is \( 12x^{-5} \).

Step 1: Identify the function form

The given function is in the form of a power function: \( y = -3x^{-4} \). The goal is to find the derivative of this function with respect to \( x \), which we'll denote as \( D_x y \) or \( \frac{dy}{dx} \).

Step 2: Apply the power rule for differentiation

The power rule for differentiation states that the derivative of \( ax^n \) is \( anx^{n-1} \). Here, \( a = -3 \) and \( n = -4 \). Applying the power rule gives:\[ \frac{dy}{dx} = -3 \cdot (-4) \cdot x^{-4-1} \]

Step 3: Simplify the expression

Carry out the multiplication and simplification:\[ \frac{dy}{dx} = 12x^{-5} \]Thus, the derivative \( D_x y \) is \( 12x^{-5} \).

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that simplifies finding the derivative of a power function. It's particularly handy because many functions can be expressed in terms of powers of variables. The rule is simple yet powerful:

• If you have a function in the form of \( ax^n \), where \( a \) and \( n \) are constants, the derivative is \( anx^{n-1} \).
• For example, the derivative of \( y = -3x^{-4} \) is found by identifying \( a = -3 \) and \( n = -4 \), and then applying the power rule.

When you apply the power rule, multiply the coefficient \( a \) by the exponent \( n \), and decrease the exponent by one. This method saves time and is often the first step in the differentiation process, making it an essential tool in your calculus toolkit.

Differentiation

Differentiation is the process of finding the derivative of a function. It's a cornerstone of calculus and helps us understand how functions change. In simpler terms, differentiation tells us the function's rate of change at any point.

• The derivative is denoted as \( D_x y \) or \( \frac{dy}{dx} \).
• In our example, we used the power rule to differentiate \( y = -3x^{-4} \), resulting in the derivative function \( 12x^{-5} \).

Finding the derivative allows us to answer questions about the slope of a tangent line at any point on a curve, and whether the function is increasing or decreasing at that point. Differentiation is also crucial for optimization problems and in physics to find velocity and acceleration from displacement-time functions.

Calculus

Calculus is a branch of mathematics that studies how things change. It's divided into two main areas: differentiation and integration. Together, they provide tools to model and analyze dynamic systems in real life, from natural sciences to economics.

• In calculus, differentiation lets us find the instantaneous rate of change of a function, which is essential in various applications.
• The process we discussed, using the power rule to find \( D_x y \) for \( y = -3x^{-4} \), is an application of differential calculus.

Through calculus, we can solve complex problems by breaking them down into simpler parts, analyzing the rate of change, and calculating accumulated quantities. It's a wide world that spans from finding the area under curves to predicting the future trajectory of a moving object.