Let's break down the problems one by one based on the image you've provided:

1. Difference Quotient:

We are given $f(x) = 5 - x^2$ and asked to calculate the difference quotient $\frac{f(x) - f(5)}{x - 5}$.

Step 1: Find $f(5)$

$f(5) = 5 - (5)^2 = 5 - 25 = -20$.

Step 2: Set up the difference quotient

$\frac{f(x) - f(5)}{x - 5} = \frac{(5 - x^2) - (-20)}{x - 5} = \frac{5 - x^2 + 20}{x - 5} = \frac{25 - x^2}{x - 5}.$

Step 3: Simplify

Notice that $25 - x^2$ can be factored as $(5 - x)(5 + x)$. Thus: $\frac{25 - x^2}{x - 5} = \frac{(5 - x)(5 + x)}{x - 5}.$ Now, since $5 - x = -(x - 5)$, we can rewrite the expression as: $\frac{-(x - 5)(5 + x)}{x - 5}.$ The $x - 5$ terms cancel out, leaving: $-(5 + x).$ Thus, the difference quotient is $-(5 + x)$.

2. Derivative $f'(5)$ using the limit definition:

The derivative at $x = 5$ is found by taking the limit: $f'(5) = \lim_{x \to 5} \frac{f(x) - f(5)}{x - 5}.$

We already know that $\frac{f(x) - f(5)}{x - 5} = -(5 + x)$. Taking the limit as $x \to 5$: $f'(5) = -(5 + 5) = -10.$ So, $f'(5) = -10$.

3. Equation of the Tangent Line:

The equation of the tangent line to $f(x)$ at $x = 5$ is given by the point-slope form: $y - f(5) = f'(5)(x - 5).$ We know that $f(5) = -20$ and $f'(5) = -10$, so the equation becomes: $y - (-20) = -10(x - 5),$ or equivalently: $y + 20 = -10(x - 5).$ Simplifying this: $y = -10(x - 5) - 20.$ Expanding the right-hand side: $y = -10x + 50 - 20 = -10x + 30.$ Thus, the equation of the tangent line is $y = -10x + 30$.

Summary:

1. Difference Quotient: $-(5 + x)$.
2. Derivative $f'(5) = -10$.
3. Equation of Tangent Line: $y = -10x + 30$.

Would you like more details on any of these steps? Here are five related questions to explore further:

1. What is the geometric interpretation of the derivative at a point?
2. How do we generalize the difference quotient for any function?
3. How can we find the second derivative of $f(x) = 5 - x^2$?
4. What does the slope of the tangent line represent in real-world applications?
5. How does the symmetry of $f(x) = 5 - x^2$ affect its derivative and tangent lines?

Tip: When simplifying difference quotients, always check if the numerator can be factored to cancel terms in the denominator.