The question on the screen asks for the values of derivatives of the function $f(x) = \frac{-2}{x}$, specifically for $f'(2)$, $f'(16)$, and $f'(-4)$.

Step 1: Find the derivative of the function $f(x) = \frac{-2}{x}$

We can differentiate $f(x)$ using the power rule. Rewrite the function as: $f(x) = -2x^{-1}$ Now, differentiate using the power rule: $f'(x) = -2 \cdot (-1) x^{-2} = \frac{2}{x^2}$

Step 2: Calculate $f'(2)$

Substitute $x = 2$ into $f'(x) = \frac{2}{x^2}$: $f'(2) = \frac{2}{2^2} = \frac{2}{4} = 0.5$

Step 3: Calculate $f'(16)$

Substitute $x = 16$ into $f'(x)$: $f'(16) = \frac{2}{16^2} = \frac{2}{256} = 0.0078125$

Step 4: Calculate $f'(-4)$

Substitute $x = -4$ into $f'(x)$: $f'(-4) = \frac{2}{(-4)^2} = \frac{2}{16} = 0.125$

Answers:

• $f'(2) = 0.5$
• $f'(16) = 0.0078$ (rounded to four decimal places)
• $f'(-4) = 0.125$

Would you like a further breakdown of any step or more details?

Here are five related questions:

1. How do you find the derivative of a function using the power rule?
2. What happens when you take the derivative of a constant multiplied by a function?
3. Can the derivative of a function be negative, and how would that affect the graph?
4. How do you interpret the second derivative of a function?
5. Why does the derivative become smaller for larger values of $x$ in the function $f(x) = \frac{-2}{x}$?

Tip: Always simplify the function into a form that is easier to differentiate (e.g., rewrite division as a power).