Find the derivative of the revenue function R(x) = 6x + 4. (This is called the marginal revenue).
Understand the Problem
The question is asking us to find the derivative of the revenue function R(x) = 6x + 4. We need to apply the power rule of differentiation to each term in the revenue function. The derivative of a constant is zero, and the derivative of 6x is 6.
Answer
$R'(x) = 6$
Answer for screen readers
$R'(x) = 6$
Steps to Solve
- Identify the revenue function
The revenue function is given as $R(x) = 6x + 4$.
- Apply the power rule to the first term
The first term is $6x$. We can write this as $6x^1$. Using the power rule, the derivative of $ax^n$ is $n \cdot ax^{n-1}$. Therefore, the derivative of $6x^1$ is $1 \cdot 6x^{1-1} = 6x^0 = 6 \cdot 1 = 6$.
- Apply the constant rule to the second term
The second term is $4$, which is a constant. The derivative of a constant is $0$.
- Combine the derivatives of both terms
The derivative of $R(x) = 6x + 4$ is the sum of the derivatives of its terms, which is $6 + 0 = 6$.
- Write the derivative of the revenue function
Thus, $R'(x) = 6$.
$R'(x) = 6$
More Information
The derivative of the revenue function, $R'(x)$, is also known as the marginal revenue. It represents the additional revenue generated by selling one more unit of a product or service.
Tips
A common mistake is to incorrectly apply the power rule or to forget that the derivative of a constant is zero. Another mistake could be not recognizing that 6x is the same as 6x^1
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