Question

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

Short answer

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

Step-by-step solution

Identify the terms of the function

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

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 \(1 * 6x^{1-1} = 6x^0 = 6\).

Derivative of constant

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

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.