Question

Compute the first four derivatives of the given function. $$p(\theta)=\theta^{4}-\theta^{3}$$

Short answer

The first four derivatives are \( p'(\theta) = 4\theta^3 - 3\theta^2 \), \( p''(\theta) = 12\theta^2 - 6\theta \), \( p'''(\theta) = 24\theta - 6 \), and \( p''''(\theta) = 24 \).

Step-by-step solution

Identify the Function

The function given is \( p(\theta) = \theta^4 - \theta^3 \). We have to find the first four derivatives of this function with respect to \( \theta \).

Compute the First Derivative

To find the first derivative, apply the power rule, which states that the derivative of \( \theta^n \) is \( n\theta^{n-1} \). Thus, the first derivative is:\[ p'(\theta) = 4\theta^3 - 3\theta^2 \]

Compute the Second Derivative

Apply the power rule to the first derivative \( p'(\theta) = 4\theta^3 - 3\theta^2 \).The second derivative is:\[ p''(\theta) = 12\theta^2 - 6\theta \]

Compute the Third Derivative

Apply the power rule to the second derivative \( p''(\theta) = 12\theta^2 - 6\theta \).The third derivative is:\[ p'''(\theta) = 24\theta - 6 \]

Compute the Fourth Derivative

Apply the power rule to the third derivative \( p'''(\theta) = 24\theta - 6 \).The fourth derivative is:\[ p''''(\theta) = 24 \]

Key concepts

Power Rule

The power rule is a fundamental concept in calculus, specifically in the topic of differentiation. It provides a quick and straightforward way of finding the derivative of polynomial functions. If you have a term like \( \theta^n \), the power rule states that you multiply by the exponent \( n \) and then decrease the exponent by one.For example, in the exercise, the term \( \theta^4 \) becomes \( 4\theta^3 \) after applying the power rule. Similarly, the term \( \theta^3 \) becomes \( 3\theta^2 \). It simplifies the process of differentiation, especially for higher-degree polynomials.

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

Understanding the power rule is essential because it forms the basis for deriving more complex functions and helps in solving calculus problems efficiently.

Calculus

Calculus is a vast branch of mathematics that deals with continuous change. It is divided into two main parts: differential calculus and integral calculus. In this exercise, we are dealing with differential calculus, which focuses on finding the rate of change of a quantity. Calculus allows us to model and solve problems involving variable rates of change, such as those found in physics, engineering, economic models, and other fields. By using calculus, we can explore how things change instantly and predict future behavior based on current trends. Some key concepts in calculus include:

• Limits - Fundamental to understanding calculus and dealing with functions as variables approach certain values.
• Derivatives - Represent the rate of change of a function and are central to differential calculus.
• Integrals - Used in integral calculus to find areas under curves and accumulate quantities.

Calculus is essential for scientific advancements and day-to-day applications because it provides a robust framework to tackle diverse dynamic systems.

Differentiation

Differentiation is the process of computing a derivative, a core operation in calculus that determines the rate at which a quantity changes. This technique is especially useful in finding slopes of curves, rates of change, and optimizing functions. To perform differentiation, like in the given exercise, you apply certain rules like the power rule, as described earlier. You take a function and calculate its derivative, which gives you a new function that represents the instantaneous rate of change of the original function. In the exercise, each step you perform—calculating the first, second, third, and fourth derivatives—involves differentiation. Each derivative provides more information:

• First derivative: Gives the slope or the rate of change at any point of the original function.
• Second derivative: Helps determine the concavity or curvature of the function (acceleration).
• Third derivative: Can offer insights into "jerk" or changes in acceleration.
• Fourth derivative: Gets progressively more abstract but can be useful in some fields like physics or engineering.

Differentiation is a powerful mathematical tool that enhances our ability to understand and predict real-world phenomena through mathematical models.