The problem in the image is as follows:

Problem: The function $y = f(x)$ has a tangent at $x = 3$ that forms a $45^\circ$ angle with the $x$-axis and passes through the point $x = -6$.

Additionally, the function $g(x)$ is defined as: $g(x) = -x \cdot f(2x - 1).$

You are tasked to find the derivative of $g(x)$ at $x = 2$, i.e., $g'(2)$.

Solution:

Step 1: Analyze the tangent line

The slope of a line forming a $45^\circ$ angle with the $x$-axis is: $m = \tan(45^\circ) = 1.$

So, at $x = 3$, the derivative of $f(x)$, $f'(3)$, is: $f'(3) = 1.$

The tangent line passes through the point $(-6, 0)$. Using the point-slope form of a line equation, the tangent line can be expressed as: $y - f(3) = m(x - 3).$ Since $m = 1$, this becomes: $y - f(3) = (x - 3).$ At $x = -6$, the tangent line crosses the $x$-axis, so: $0 - f(3) = -6 - 3 \implies f(3) = 9.$

Thus: $f(3) = 9, \quad f'(3) = 1.$

Step 2: Differentiate $g(x)$

The function $g(x)$ is given as: $g(x) = -x \cdot f(2x - 1).$

To find $g'(x)$, use the product rule: $g'(x) = \frac{d}{dx} \big[-x \cdot f(2x - 1)\big].$ Applying the product rule: $g'(x) = (-1) \cdot f(2x - 1) + (-x) \cdot \frac{d}{dx} \big[f(2x - 1)\big].$

The derivative of $f(2x - 1)$ involves the chain rule: $\frac{d}{dx} \big[f(2x - 1)\big] = f'(2x - 1) \cdot \frac{d}{dx}(2x - 1) = f'(2x - 1) \cdot 2.$

Thus: $g'(x) = -f(2x - 1) - 2x \cdot f'(2x - 1).$

Step 3: Evaluate $g'(2)$

Substitute $x = 2$ into $g'(x)$: $g'(2) = -f(2(2) - 1) - 2(2) \cdot f'(2(2) - 1).$

Simplify: $g'(2) = -f(4 - 1) - 4 \cdot f'(4 - 1).$ $g'(2) = -f(3) - 4 \cdot f'(3).$

From Step 1, we know: $f(3) = 9, \quad f'(3) = 1.$

Substitute these values: $g'(2) = -9 - 4 \cdot 1 = -9 - 4 = -13.$

Final Answer: $g'(2) = -13.$

Would you like me to elaborate on any part of the solution?

5 Related Questions:

1. How do we compute derivatives using the chain rule in general cases?
2. What is the product rule for differentiation, and when is it applied?
3. How does the slope of a tangent line relate to the derivative of a function?
4. How can we verify the slope of a line given two points?
5. What is the importance of defining composite functions like $f(2x - 1)$?

Tip: Always ensure that the slope of a tangent is consistent with the angle provided, as small errors here can lead to incorrect solutions.