To find the linearization of the function $f(x) = \frac{1}{5x + 4}$ at $x = -1$, we follow these steps:

Step 1: Find $f(-1)$

Substitute $x = -1$ into the function to find the value of the function at that point.

$f(-1) = \frac{1}{5(-1) + 4} = \frac{1}{-5 + 4} = \frac{1}{-1} = -1$

So, $f(-1) = -1$.

Step 2: Find the derivative of $f(x)$

To find the derivative of $f(x) = \frac{1}{5x + 4}$, we apply the chain rule. Let $g(x) = 5x + 4$, so $f(x) = \frac{1}{g(x)}$. The derivative of $f(x)$ is:

$f'(x) = -\frac{1}{(g(x))^2} \cdot g'(x)$

Where $g'(x) = 5$, so:

$f'(x) = -\frac{1}{(5x + 4)^2} \cdot 5 = -\frac{5}{(5x + 4)^2}$

Step 3: Find $f'(-1)$

Substitute $x = -1$ into the derivative to find $f'(-1)$:

$f'(-1) = -\frac{5}{(5(-1) + 4)^2} = -\frac{5}{(-5 + 4)^2} = -\frac{5}{(-1)^2} = -\frac{5}{1} = -5$

So, $f'(-1) = -5$.

Step 4: Write the linear approximation (Linearization)

The formula for the linearization of a function $f(x)$ at a point $x = a$ is given by:

$L(x) = f(a) + f'(a)(x - a)$

Substitute $a = -1$, $f(-1) = -1$, and $f'(-1) = -5$:

$L(x) = -1 + (-5)(x - (-1)) = -1 - 5(x + 1)$

Simplify the expression:

$L(x) = -1 - 5(x + 1) = -1 - 5x - 5 = -6 - 5x$

Final Answer:

The linearization of $f(x) = \frac{1}{5x + 4}$ at $x = -1$ is:

$L(x) = -5x - 6$

---

Would you like further details or have any questions on this process?

Here are some additional questions for exploration:

1. What is the geometric interpretation of linearization at a given point?
2. How would you use linear approximation in practical problems?
3. Can you use linearization for functions that are not differentiable at the point of interest?
4. How do you verify the accuracy of a linear approximation?
5. What are some common applications of linearization in physics and engineering?

Tip: Always check if the function is differentiable at the point of interest before applying linearization!