To solve this problem, we'll proceed in two steps:

1. Find $g'(5)$ using the formal definition of the derivative at a point.
2. Write the equation of the tangent line at $x = 5$ in point-slope form.

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Step 1: Find $g'(5)$ using the formal definition of the derivative

The function given is:

$g(x) = \sqrt{2x - 1}$

The formal definition of the derivative at a point $x = a$ is:

$g'(a) = \lim_{h \to 0} \frac{g(a + h) - g(a)}{h}$

In our case, $a = 5$. So, we'll compute $g'(5)$.

Step 1.1: Compute $g(5)$

$g(5) = \sqrt{2(5) - 1} = \sqrt{10 - 1} = \sqrt{9} = 3$

Step 1.2: Apply the definition of the derivative

Now we use the definition of the derivative:

$g'(5) = \lim_{h \to 0} \frac{g(5 + h) - g(5)}{h}$

Substitute $g(x) = \sqrt{2x - 1}$:

$g'(5) = \lim_{h \to 0} \frac{\sqrt{2(5 + h) - 1} - 3}{h}$ $g'(5) = \lim_{h \to 0} \frac{\sqrt{10 + 2h - 1} - 3}{h}$ $g'(5) = \lim_{h \to 0} \frac{\sqrt{9 + 2h} - 3}{h}$

To evaluate this limit, we rationalize the numerator by multiplying the top and bottom by the conjugate:

$g'(5) = \lim_{h \to 0} \frac{\left( \sqrt{9 + 2h} - 3 \right)}{h} \times \frac{\left( \sqrt{9 + 2h} + 3 \right)}{\left( \sqrt{9 + 2h} + 3 \right)}$ $g'(5) = \lim_{h \to 0} \frac{(9 + 2h) - 9}{h \left( \sqrt{9 + 2h} + 3 \right)}$ $g'(5) = \lim_{h \to 0} \frac{2h}{h \left( \sqrt{9 + 2h} + 3 \right)}$

Simplify:

$g'(5) = \lim_{h \to 0} \frac{2}{\sqrt{9 + 2h} + 3}$

As $h \to 0$, $\sqrt{9 + 2h} \to \sqrt{9} = 3$. Therefore, the limit becomes:

$g'(5) = \frac{2}{3 + 3} = \frac{2}{6} = \frac{1}{3}$

So, $g'(5) = \frac{1}{3}$.

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Step 2: Write the equation of the tangent line

The equation of the tangent line at $x = 5$ is given by the point-slope form:

$y - y_1 = m(x - x_1)$

Where:

• $m = g'(5) = \frac{1}{3}$ is the slope of the tangent line,
• $(x_1, y_1) = (5, g(5)) = (5, 3)$ is the point of tangency.

Substitute these values into the point-slope form:

$y - 3 = \frac{1}{3}(x - 5)$

Thus, the equation of the tangent line is:

$y - 3 = \frac{1}{3}(x - 5)$

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Let me know if you'd like more details or if you have any questions.

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5 Related Questions:

1. How is the formal definition of a derivative applied to different functions?
2. How would the process change if $g(x)$ had a more complex form?
3. Can you find $g'(x)$ for a general $x$ in this function?
4. What is the significance of the tangent line at a particular point?
5. How do we determine the limit if the numerator involves more complicated expressions?

Tip:

When rationalizing expressions involving square roots, multiplying by the conjugate can simplify limits and help you evaluate derivatives.