Problem 12 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(x)=6 x+3 $$

Step-by-Step Solution

Answer

The derivative of the given function \(g(x) = 6x + 3\) is \(g'(x) = 6\).

Step 1: Identify the terms of the function

The function \(g(x) = 6x + 3\) can be broken down into two separate terms: \(6x\) and \(3\).

Step 2: Apply power rule

The power rule states that the derivative of \(ax^n\) is \(nax^{n-1}\). The term \(6x\) can be considered as \(6x^1\), so applying the power rule we get \(1 * 6x^{1-1} = 6x^0 = 6\).

Step 3: Derivative of constant

The derivative of a constant is zero. So, the derivative of the constant term \(3\) is zero.

Step 4: Combine the results

Combine the derivatives of each term to find the derivative of the whole function. Therefore, \(g'(x) = 6 + 0 = 6\).

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus used to find the derivative of a term in the form of \( ax^n \), where \( a \) is a constant, and \( n \) is the power of \( x \). To use the power rule, you follow a simple process:

• Bring down the exponent as a coefficient.
• Multiply it by the original coefficient \( a \).
• Subtract one from the original exponent \( n \).

For example, if we apply this rule to the term \( 6x^1 \), the derivative becomes \( 1 \times 6 \times x^{1-1} = 6x^0 = 6 \).
This tells us that the rate of change of \( 6x \) with respect to \( x \) is constant at 6. Remember, when using the power rule, any variable should be expressed as a power of \( x \), even if that is simply \( x^1 \).
By mastering the power rule, you can quickly find derivatives of polynomial functions, simplifying the process of differentiation.

Derivative of a Constant

The derivative of a constant is an important and straightforward concept in calculus. A constant is any number without a variable attached to it. When you take the derivative of a constant, the result is always zero. This is because constants do not change, and thus, have no rate of change.
Consider the function \( g(x) = 6x + 3 \). The constant term here is \( 3 \). When finding its derivative, remember:

• Constants are unaffected by changes in \( x \).
• The derivative of \( 3 \) (or any constant) is \( 0 \).

This reflects the idea that adding or subtracting a constant to a function does not affect its slope or the rate at which it changes. It remains horizontal in terms of differentiation.
Recognizing the derivative of a constant helps to simplify complex expressions quickly and is a handy tool in your calculus toolkit.

Derivative

A derivative, in calculus, represents the rate of change of a function concerning its variable. It is a fundamental concept that underpins much of calculus. When you calculate a derivative, you are essentially finding the slope of the tangent line at any point on the function.
For the function \( g(x) = 6x + 3 \), the derivative tells us how \( g(x) \) changes as \( x \) changes. To find this:

• Use the power rule on the \( 6x \) term to get 6.
• Apply the rule of derivatives for constants to\( 3 \), which yields 0.
• Combine results to express the overall rate of change.

Thus, the derivative \( g'(x) = 6 \) implies that for every unit increase in \( x \), the function increases steadily by 6 units.
Understanding derivatives is crucial for analyzing and interpreting the behavior of functions in various applications, from engineering to economics.