Problem 32 - Technical Mathematics with Calculus

Question

Find the derivative. $$D(3 x+2)$$

Step-by-Step Solution

Answer

3

Step 1: Recognize the Function

The function given for differentiation is a linear function, in the form of f(x) = ax + b, where a and b are constants.

Step 2: Apply the Power Rule

Use the power rule for differentiation, which states that the derivative of x to the power of n is nx^(n-1). Here, x is to the power of 1, so the derivative of x is 1.

Step 3: Differentiate the Constant Terms

Remember that the derivative of any constant is zero. Since there are no variables attached to the constants in the function, their derivative will be zero.

Step 4: Compute the Derivative

Combine the derivatives of the individual terms by multiplying the constants by the derivative of x, and summing them up. Since the derivative of 3x is 3 and the derivative of 2 is zero, the result is the derivative of the function.

Key Concepts

Linear Function Differentiation

Differentiating a linear function is a foundational skill in calculus. A linear function is an algebraic equation in which each term is either a constant or the product of a constant and the first power of a single variable. Linear functions take the general form of f(x) = ax + b, where a and b are constants, and x represents the variable.

When finding the derivative of such functions, you'll notice that they change at a constant rate, which means their steepness or slope is consistent. This is why the derivative of a linear function is simply the coefficient of x. For the function D(3x + 2), the rate of change is 3, hence the derivative is 3. It reflects the fact that for every one unit of change in x, y changes by three units. Understanding this concept ensures that students can quickly identify and differentiate linear functions.

Power Rule for Differentiation

The power rule is a basic yet powerful tool used to differentiate functions of the form x^n, where n is any real number. The rule states that the derivative of x^n is nx^(n-1). This means that you bring down the exponent as a coefficient in front of the variable and then subtract one from the exponent.

For instance, in our exercise, the term 3x is essentially 3x^1. Applying the power rule, we see that the derivative is 1*3x^(1-1) which simplifies to 3. Remember that x^0 equals 1, and hence it disappears in the final derivative, leaving us with the coefficient 3 as the derivative. When students grasp the mechanics of the power rule, they can differentiate a wide variety of polynomial functions with confidence and ease.

Differentiating Constant Terms

In the realm of calculus, differentiating constant terms is as straightforward as it gets. Simply put, the derivative of any constant is zero. Why is that? A constant does not change, so its rate of change is always zero. Therefore, when we differentiate a function that includes a constant term, that term disappears in the derivative.

In our exercise example, 2 is the constant term in the linear function 3x + 2. According to the rules of differentiation, the derivative of 2 is 0. It's essential for students to recognize constant terms and understand that they do not contribute to the derivative of a function. This knowledge simplifies calculus problems considerably by reducing the number of terms that need to be differentiated. As you apply this concept consistently, the process of finding derivatives becomes more intuitive and efficient.