Problem 28 - Technical Mathematics with Calculus

Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Sum. $$y=x^{3}-x^{3 / 2}+3 x$$

Step-by-Step Solution

Answer

The derivative of the function \(y = x^{3} - x^{3 / 2} + 3x\) is \(y' = 3x^2 - \frac{3}{2}x^{1/2} + 3\).

Step 1: Identify the components of the function

Break down the function into its individual terms: the first term is a cubic function, the second term is a function to the 3/2 power, and the third term is a linear function.

Step 2: Apply the power rule individually to each term

Use the power rule for differentiation, which states that the derivative of a function of the form \(x^n\) is \(nx^{n-1}\), to find the derivatives of the individual terms.

Step 3: Differentiate the first term

The derivative of \(x^3\) is \(3x^2\).

Step 4: Differentiate the second term

The derivative of \(-x^{3/2}\) is \(-\frac{3}{2}x^{3/2 - 1} = -\frac{3}{2}x^{1/2}\).

Step 5: Differentiate the third term

The derivative of \(3x\) is \(3\).

Step 6: Combine the derivatives

Combine the derivatives of all individual terms to obtain the derivative of the entire function.

Key Concepts

Power Rule Differentiation

The power rule is a fundamental concept in calculus that streamlines the process of differentiation when dealing with polynomial terms. Typically, a polynomial function will consist of one or more terms, each with a variable raised to an exponent. To find the derivative of a term in the form of \(x^n\), where n is a constant exponent, the power rule instructs us to multiply the term by the exponent and then decrease the exponent by one.

For example, applying the power rule to differentiate \(x^3\), you would multiply the term by 3 (the exponent) and then subtract one from the exponent, yielding \(3x^2\) as the derivative. It's an elegantly simple yet powerful technique used in differentiation.

Derivative of a Sum

When faced with finding the derivative of a function that is the sum of several terms, calculus offers a straightforward approach: differentiate each term individually and then combine the results. This is based on the principle that the derivative of a sum is the sum of the derivatives. In the exercise presented, the function \(y=x^3-x^{3/2}+3x\) is composed of three distinct terms.

Following the derivative of a sum rule, we separately apply differentiation to \(x^3\), \(x^{3/2}\), and \(3x\), yielding \(3x^2\), \(\frac{3}{2}x^{1/2}\), and \(3\), respectively. After finding the derivatives of each term, we simply add them together (taking into consideration any negative signs) to get the derivative of the entire function.

Calculus

Calculus, the branch of mathematics that studies continuous change, is divided into two main parts: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. It's used widely for optimizing problems, motion analysis, and in the field of physics, among other applications.

The power rule and the derivative of a sum are essential tools within differential calculus and are applied to various functions to find their derivatives with respect to an independent variable, frequently represented as \(x\). Calculus as a whole presents a framework not just for solving mathematical problems but also for understanding patterns and systems that evolve continuously over time.