Problem 23 - 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. $$y=3-2 x$$

Step-by-Step Solution

Answer

The derivative of the function \(y = 3 - 2x\) is \(-2\).

Step 1: Identify the Function

Begin by identifying the function for which you want to find the derivative. In this case, the function is given by the formula: \(y = 3 - 2x\).

Step 2: Apply the Sum Rule

The sum rule in differentiation states that the derivative of a sum is the sum of the derivatives. Since our function is a sum (difference) of two terms, we can differentiate each term individually. The derivative of a constant is 0, and the derivative of \(-2x\) with respect to \(x\) is simply its coefficient \(-2\).

Step 3: Combine the Derivatives

After applying the sum rule, combine the derivatives of the individual terms. The derivative of the constant term (3) is 0, and the derivative of \(-2x\) is \(-2\). So, the derivative of the entire function is simply \(-2\).

Step 4: Verify with a Calculator (Optional)

Use a calculator to verify the answer if desired. Input the function into the calculator's differentiation tool to confirm that the derivative is indeed \(-2\).

Key Concepts

Sum Rule in Differentiation

When we're faced with finding the derivative of a function that's a sum of several terms, we can use a helpful trick known as the sum rule in differentiation. This rule states that the derivative of a sum is simply the sum of the derivatives of its parts. Think of it like breaking down a big job into smaller, more manageable tasks.

In the exercise provided, the function is broken down into two parts: a constant (\(3\)) and a variable term multiplied by a constant ((-2x). According to the sum rule, we differentiate these parts separately. Differentiation is like a machine that processes each term individually. For the constant, this machine returns a zero because constants don't change, they don't have a rate of change, which is what derivatives measure. For the variable term, the machine spits out the coefficient (-2) because that's the rate at which that term changes with respect to the variable.

In essence, when faced with a function like (y = 3 - 2x), the sum rule makes our life easier by allowing us to differentiate piece by piece: the derivative of (3) (a constant) is (0) and the derivative of (-2x) with respect to (x) is (-2). When we combine these results, we get the overall derivative of the function, which is simply (-2).

Differentiating Constants

A constant is like that unchanging friend who's always there, reliable and steady. In the world of mathematics, constants are numbers that don't depend on the value of a variable—they stand on their own with unwavering value. So, what happens when we encounter them during differentiation? It's simple: they vanish, like a ghost in mathematics!

This disappearing act happens because differentiation is concerned with how things change, and constants, well, they don't change at all. Hence, when we take the derivative of a constant term in an algebraic expression, the result is zero. It's like asking how much a rock grows each year – the answer is it doesn’t. In our exercise, when we differentiated the term (3), a constant, its derivative was (0) because it has no rate of change. This principle helps clean up our work by removing static parts that don't contribute to the overall variation of the function.

Verifying Derivatives with a Calculator

While mastering the rules of differentiation is important for developing a deep understanding of calculus, sometimes you just want to make sure your answer is on the right track. That's where calculators equipped with differentiation capabilities can be a real sanity check. These handy tools allow us to input a function and quickly obtain its derivative, serving as a verification against our manual computations.

In our textbook example, after calculating the derivative to be (-2), we could utilize a calculator as a means to double-check. By entering the function (y = 3 - 2x) into the calculator and using its differentiation function, we can confirm that our hand-calculated derivative matches the calculator’s output. This step is like asking for a second opinion; it reassures us that our process and result align with mathematical truths. However, it's crucial to understand the process behind differentiation to truly grasp the subject and not just rely on technology. Think of calculators as your mathematical sidekick, there to support, but not to overshadow your own problem-solving abilities.