Problem 11 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ f(x)=x+11 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=x+11\) is \(1\).

Step 1: Identify the Function

The function given is \(f(x)=x+11\). In this function, there are two terms - \(x\) and \(11\).

Step 2: Apply the Derivative Rule

Now, it's time to apply the power rule of differentiation. According to this rule, the derivative of the term \(x\) with respect to \(x\) is \(1\) and also the derivative of any constant, such as \(11\) in this case, is \(0\).

Step 3: Find the Derivative

Using the rules of differentiation, the derivative of the function \(f(x)=x+11\) is: \(f'(x) = 1 + 0 = 1\).

Key Concepts

Power Rule in Differentiation

The power rule is one of the basic tools in differentiation. It is especially useful for functions where terms involve variables raised to an exponent. The power rule states that if you have a function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative, or \(f'(x)\), is found by multiplying the coefficient by the exponent and reducing the exponent by one.In simpler terms, the power rule can be broken down into the following steps:

• Multiply the current power of the variable with its coefficient.
• Subtract 1 from the current power.

For example, for \(f(x) = x^3\), the derivative \(f'(x)\) would be \(3x^{3-1} = 3x^2\). In our specific exercise, when differentiating \(f(x) = x + 11\), the \(x\) term is essentially \(x^1\), making the derivative calculation straightforward as it applies the power rule resulting in simply \(1\).

Derivative of a Constant

A crucial part of understanding derivatives is grasping the concept of the derivative of a constant. In differentiation, constants are defined as numbers without variables, such as \(11\) in our exercise.The rule is straightforward: the derivative of a constant is always \(0\). This means that if you are taking the derivative of a function such as \(f(x) = b\) where \(b\) is a constant, the derivative is simply \(f'(x) = 0\).

• Constant terms do not change as you vary the variable they accompany.
• This makes their rate of change, or derivative, identically zero.

In the exercise with \(f(x) = x + 11\), the 11 does not change as \(x\) changes, hence its derivative is 0. The only term left to consider in the differentiation is \(x\), leading to a simple answer.

Calculus Problem Solving

Solving calculus problems often involves applying a series of rules and principles of derivatives systematically. When facing a function, the first step is recognizing the types of terms involved and which rules to apply, as in the given problem where we identify both a variable term \(x\) and a constant \(11\).After identifying the terms, utilize relevant differentiation rules efficiently:

• Apply the power rule to variable terms.
• Use the constant rule for numbers without variables.

In the original exercise, using these straightforward rules simplifies the calculation significantly. Understanding these processes in calculus allows you to solve these types of problems with confidence and accuracy. You can apply these same techniques to more complex functions, knowing that breaking it down into smaller parts and applying the rules methodically will often lead you to the correct solution.