Final answer:
The derivative of the function f(x) = x^2 − 9x + 6 at x=5 is 1. To find the derivative at any point, calculate the general derivative 2x − 9, then substitute the specific value or variable.
Step-by-step explanation:
The derivative of the given function f(x) = x^2 − 9x + 6 must be found at specific points: 5 and a variable a. The first step is to calculate the general derivative of the function using basic differentiation rules.
The derivative of f(x) = x^2 − 9x + 6 at the point (a) x=5 is f'(5) = 2(5) − 9 = 1, and (b) at a general point x=a is f'(a) = 2a − 9.
To find the derivative of the function, we apply the power rule to each term separate− for the term x^2, we get 2x; for −9x, we get −9, and the derivative of a constant is 0. Therefore, f'(x) = 2x − 9. Now, we can find the derivative at a specific point by substituting the value into the derivative function. For point (a) we substitute x with 5, resulting in f'(5) = 2(5) − 9 = 10 − 9 = 1. For point (b) with a general point x=a, we substitute x with a, giving us f'(a) = 2a − 9. Thus, we obtain the slopes of the tangent lines of the function at these specific points.