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"3. Write the equation of a tangent line to the function evaluated at the given point. f(x) = 2 ln(x) at x = 3. What is the equation of the tangent line in y = mx + b form?"

Solution:

To find the equation of the tangent line, we need to follow these steps:

1. Find the derivative of the function $f(x) = 2 \ln(x)$ to determine the slope of the tangent line.

   The derivative $f'(x)$ represents the slope of the tangent line at any point $x$.

   $f'(x) = \frac{d}{dx} [2 \ln(x)] = \frac{2}{x}$

2. Evaluate the derivative at $x = 3$ to find the slope of the tangent line at that specific point.

   $f'(3) = \frac{2}{3}$

   So, the slope $m$ of the tangent line at $x = 3$ is $\frac{2}{3}$.

3. Find the function value at $x = 3$ to determine the point on the curve where the tangent line touches.

   $f(3) = 2 \ln(3)$

   Therefore, the point is $(3, 2\ln(3))$.

4. Use the point-slope form of a line to find the equation of the tangent line. The point-slope form is given by:

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

   Substituting $m = \frac{2}{3}$, $x_1 = 3$, and $y_1 = 2 \ln(3)$:

   $y - 2 \ln(3) = \frac{2}{3}(x - 3)$

5. Solve for $y$ to get the equation in the form $y = mx + b$:

   $y = \frac{2}{3}x + 2 \ln(3) - 2$

Thus, the equation of the tangent line in $y = mx + b$ form is:

$y = \frac{2}{3}x + 2 \ln(3) - 2$

Do you want details or have any questions?

Here are five related questions to consider:

1. How do you find the derivative of more complex logarithmic functions?
2. What is the significance of the point-slope form of a line?
3. How would the tangent line change if the point were different, say at $x = 1$?
4. What are some real-world applications of finding the tangent line to a curve?
5. Can you explain why the tangent line is an approximation of the function near the point of tangency?

Tip: Understanding how to find the derivative is key to solving problems involving the slope of a tangent line. Practice differentiating various types of functions to improve your skills.