If $f(x) = \frac{3x + 3}{5x + 6}$, Find:1. $f^{\prime}(x) =$2. $f^{\prime}(4) =$

If $f(x) = \frac{3x + 3}{5x + 6}$, find: 1. $f^{\prime}(x) =$2. $f^{\prime}(4) =$

In this article, we will explore the concept of finding the derivative of a given function and then evaluating it at a specific point. The function in question is $f(x) = \frac{3x + 3}{5x + 6}$. We will first find the derivative of this function, denoted as $f^{\prime}(x)$, and then evaluate it at $x = 4$.

To find the derivative of a function, we can use the quotient rule, which states that if we have a function of the form $f(x) = \frac{g(x)}{h(x)}$, then the derivative of this function is given by:

$$f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{(h(x))^2}$$

In our case, we have $g(x) = 3x + 3$ and $h(x) = 5x + 6$. We can find the derivatives of these functions as follows:

$$g^{\prime}(x) = 3$$

$$h^{\prime}(x) = 5$$

Now, we can plug these values into the quotient rule formula:

$$f^{\prime}(x) = \frac{(5x + 6)(3) - (3x + 3)(5)}{(5x + 6)^2}$$

Simplifying this expression, we get:

$$f^{\prime}(x) = \frac{15x + 18 - 15x - 15}{(5x + 6)^2}$$

$$f^{\prime}(x) = \frac{3}{(5x + 6)^2}$$

$$

Now that we have found the derivative of the function, we can evaluate it at $x = 4$. Plugging $x = 4$ into the derivative, we get:

$$f^{\prime}(4) = \frac{3}{(5(4) + 6)^2}$$

$$f^{\prime}(4) = \frac{3}{(26)^2}$$

$$f^{\prime}(4) = \frac{3}{676}$$

In this article, we found the derivative of the function $f(x) = \frac{3x + 3}{5x + 6}$ and then evaluated it at $x = 4$. We used the quotient rule to find the derivative and then simplified the expression to get the final answer. The derivative of the function is $f^{\prime}(x) = \frac{3}{(5x + 6)^2}$, and the value of the derivative at $x = 4$ is $f^{\prime}(4) = \frac{3}{676}$.

Here is a step-by-step solution to the problem:

1. Find the derivatives of $g(x) = 3x + 3$ and $h(x) = 5x + 6$.
2. Plug these values into the quotient rule formula.
3. Simplify the expression to get the final answer.
4. Evaluate the derivative at $x = 4$.

Here are some common mistakes to avoid when finding the derivative of a function:

• Not using the quotient rule when the function is in the form $f(x) = \frac{g(x)}{h(x)}$.
• Not simplifying the expression after finding the derivative.
• Not evaluating the derivative at the correct point.

The concept of finding the derivative of a function has many real-world applications. For example:

• In physics, the derivative of a function can be used to find the velocity and acceleration of an object.
• In economics, the derivative of a function can be used to find the marginal cost and marginal revenue of a product.
• In engineering, the derivative of a function can be used to find the slope of a curve and the rate of change of a quantity.

In our previous article, we explored the concept of finding the derivative of a function and evaluating it at a specific point. We used the quotient rule to find the derivative of the function $f(x) = \frac{3x + 3}{5x + 6}$ and then evaluated it at $x = 4$. In this article, we will answer some common questions related to finding the derivative of a function.

Q: What is the quotient rule?

A: The quotient rule is a formula for finding the derivative of a function that is in the form $f(x) = \frac{g(x)}{h(x)}$. The quotient rule states that the derivative of this function is given by:

$$f^{\prime}(x) = \frac{h(x)g^{\prime}(x) - g(x)h^{\prime}(x)}{(h(x))^2}$$

Q: How do I find the derivative of a function using the quotient rule?

A: To find the derivative of a function using the quotient rule, you need to follow these steps:

1. Identify the functions $g(x)$ and $h(x)$.
2. Find the derivatives of $g(x)$ and $h(x)$.
3. Plug these values into the quotient rule formula.
4. Simplify the expression to get the final answer.

Q: What is the difference between the quotient rule and the product rule?

A: The quotient rule and the product rule are two different formulas for finding the derivative of a function. The quotient rule is used when the function is in the form $f(x) = \frac{g(x)}{h(x)}$, while the product rule is used when the function is in the form $f(x) = g(x)h(x)$. The product rule states that the derivative of this function is given by:

$$f^{\prime}(x) = g^{\prime}(x)h(x) + g(x)h^{\prime}(x)$$

Q: How do I evaluate the derivative of a function at a specific point?

A: To evaluate the derivative of a function at a specific point, you need to plug the value of the point into the derivative. For example, if we want to evaluate the derivative of the function $f(x) = \frac{3x + 3}{5x + 6}$ at $x = 4$, we would plug $x = 4$ into the derivative:

$$f^{\prime}(4) = \frac{3}{(5(4) + 6)^2}$$

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Some common mistakes to avoid when finding the derivative of a function include:

• Not using the correct formula for the derivative.
• Not simplifying the expression after finding the derivative.
• Not evaluating the derivative at the correct point.

Q: What are some real-world applications of finding the derivative of a function?

A: The concept of finding the derivative of a function has many real-world applications, including:

• In physics, the derivative of a function can be used to find the velocity and acceleration of an object.
• In economics, the derivative of a function can be used to find the marginal cost and marginal revenue of a product.
• In engineering, the derivative of a function can be used to find the slope of a curve and the rate of change of a quantity.

In conclusion, finding the derivative of a function and evaluating it at a specific point is an important concept in mathematics. The quotient rule is a useful tool for finding the derivative of a function, and the concept of finding the derivative has many real-world applications. By following the steps outlined in this article, you can find the derivative of a function and evaluate it at a specific point.

For more information on finding the derivative of a function, you can consult the following resources:

• Calculus textbooks
• Online tutorials and videos
• Math websites and forums

Here are some practice problems to help you practice finding the derivative of a function:

1. Find the derivative of the function $f(x) = \frac{2x + 1}{3x - 2}$.
2. Evaluate the derivative of the function $f(x) = \frac{3x + 2}{4x - 1}$ at $x = 3$.
3. Find the derivative of the function $f(x) = (2x + 1)(3x - 2)$.

Here are the answers to the practice problems:

1. $f^{\prime}(x) = \frac{3(3x - 2) - (2x + 1)(3)}{(3x - 2)^2}$
2. $f^{\prime}(3) = \frac{3(4) - (5)(3)}{(4)^2} = \frac{12 - 15}{16} = -\frac{3}{16}$
3. $f^{\prime}(x) = 6x - 6$