To differentiate $f(x) = 5x - 4$ at $x = 2$ using the first principle of differentiation, we follow these steps:

First Principle of Differentiation Formula:

The derivative of a function $f(x)$ at a point $x = a$ is given by: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Step-by-Step Solution:

1. Substitute the function and point:

   • The function is $f(x) = 5x - 4$.
   • We need to find the derivative at $x = 2$, so $a = 2$.

2. Apply the first principle formula: $f'(2) = \lim_{h \to 0} \frac{f(2 + h) - f(2)}{h}$

3. Evaluate $f(2 + h)$ and $f(2)$:

   • $f(2 + h) = 5(2 + h) - 4 = 10 + 5h - 4 = 6 + 5h$
   • $f(2) = 5(2) - 4 = 10 - 4 = 6$

4. Substitute into the formula: $f'(2) = \lim_{h \to 0} \frac{(6 + 5h) - 6}{h} = \lim_{h \to 0} \frac{5h}{h}$

5. Simplify the expression: $f'(2) = \lim_{h \to 0} 5 = 5$

Final Answer:

The derivative of $f(x) = 5x - 4$ at $x = 2$ using the first principle of differentiation is $\boxed{5}$.

Would you like further details or have any questions?

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Here are 5 related questions:

1. How is the derivative of a constant function computed?
2. Can the first principle of differentiation be used for higher-order polynomials?
3. What is the geometric interpretation of the derivative of a linear function?
4. How would you compute the derivative of a quadratic function at a point?
5. What is the difference between differentiation from first principles and using standard rules?

Tip: The first principle of differentiation works for all differentiable functions but is particularly straightforward for linear functions.