Final answer:
To compute f′(9) using the limit definition f′(9)=, we need to find the derivative of f(x) = √73−x. By following the steps of the limit definition of the derivative and simplifying the expression, we can find the value of f′(9). However, the question also asks for the equation of the tangent line at x = 9, which was not provided.
Step-by-step explanation:
Mathematics - High School
To find f′(9), we need to compute the derivative of the function f(x) = √73−x using the limit definition of the derivative. Let's calculate it step-by-step:
- Write the limit definition of the derivative: f'(x) = lim(h→0) [(f(x+h) - f(x))/h]
- Substitute x = 9 into the definition: f'(9) = lim(h→0) [(f(9+h) - f(9))/h]
- Expand f(x) at x = 9: f(9) = √(73 - 9) = √64 = 8
- Compute f(9+h): f(9+h) = √(73 - (9+h)) = √(64 - h)
- Substitute the expressions back into the limit definition and simplify: f'(9) = lim(h→0) [√(64 - h) - 8]/h
At this point, we can choose to rationalize the numerator or use L'Hôpital's rule to simplify the expression further. Finally, after evaluating the limit, we can find f′(9). However, the given equation of the tangent line is missing, so we cannot provide the equation of the tangent line at x = 9.