Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems } 1-44, \text { find } D_{x} y \text { using the rules of this section. } $$ $$ y=-3 x^{-4} $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Differentiation Rule

We are given the function \( y = -3x^{-4} \). To find the derivative \( D_x y \), we need to use the power rule for differentiation, which states \( \frac{d}{dx}[x^n] = nx^{n-1} \) for any real number \( n \).

Step 2: Apply the Power Rule

Applying the power rule, where \( n = -4 \), we take the derivative: \( \frac{d}{dx}[-3x^{-4}] = -3 \cdot (-4) \cdot x^{-4-1} \). Simplifying, this becomes \( 12x^{-5} \).

Step 3: Write the Final Derivative

Thus, the derivative of \( y = -3x^{-4} \) is \( D_x y = 12x^{-5} \).

Key Concepts

power rule

The power rule is a fundamental concept in calculus used to differentiate functions. It is especially handy for functions that are polynomials or those involving terms with variables raised to a power. The rule states that if you have a term in the form of \( x^n \), where \( n \) is any real number, the derivative of that term is given by \( nx^{n-1} \). This means you're essentially multiplying by the exponent and reducing the exponent by one.

When applying the power rule, consider these steps:

• Identify the exponent \( n \).
• Multiply the entire term by the exponent \( n \).
• Subtract one from the exponent.

In the original exercise, we used the power rule to find the derivative of \( y = -3x^{-4} \). Here, \( n = -4 \). By applying the rule, we multiplied \(-3\) by \(-4\) to get \(12\) and then decreased the exponent by one, resulting in a new term with \( x^{-5} \).

derivative

A derivative represents the rate at which a function is changing at any given point. It is one of the core concepts in calculus and provides insights into understanding how functions behave.

Think of a derivative as the function's instantaneous rate of change, similar to how speed is the rate of change of position. For a given function \( y = f(x) \), the derivative \( f'(x) \) is often referred to as \( \frac{dy}{dx} \). This notation emphasizes the change in \( y \) with respect to the change in \( x \).

Understanding derivatives helps in:

• Finding slopes of tangent lines to curves.
• Determining which directions functions are increasing or decreasing.
• Solving problems related to motion, like velocity and acceleration.

In the exercise, we calculated the derivative \( D_x y = 12x^{-5} \), which tells us how \( y \) changes in response to changes in \( x \) for the function \( y = -3x^{-4} \). This new function captures the slope at each point along the curve of \( y \).

calculus

Calculus, at its core, is the study of change. It offers tools and techniques to deal with continually changing situations and is divided into two main branches: differential calculus and integral calculus.

Differential calculus focuses on the concept of a derivative, which allows us to explain and predict how quantities change over time or under certain conditions. Integration, on the other hand, is about accumulation and areas.

Some key aspects of calculus include:

• Describing how entities evolve dynamically.
• Calculating areas under curves, which is the core function of integrals.
• Finding maxima and minima of functions, which involve derivatives.

Our exercise tapped into the fascinating world of differential calculus by using the power rule to find the derivative of a function. This approach allowed us to simplify the problem and find meaningful results that describe how the function behaves.