Find the function f given that the slope of the tangent line to the graph of f at any point

Question

Find the function f given that the slope of the tangent line to the graph of f at any point (x, f(x))is f '(x) and that the graph of f passes through the given point. (Remember to use absolute values where appropriate.)

f '(x) =(9/x)+ 1; (1, 8)

Answer 1

To find f(x), we need to integrate f'(x). We have:

f(x) = ∫ [(9/x) + 1]dx = 9ln |x| + x + C

Now, we need to find the specific value of C. We know that f(1) = 8, so we have:

8 = 9ln 1 + 1 + C = 1 + C

C = 7

Therefore, f(x) = 9ln |x| + x + 7