Find g'(5) given that f(5) = -6, f'(5) = 8, and g(x) = 9x + 7 f(x). (Round your answer to four decimal places.)
Solution:
To find g'(5), we need to differentiate the function g(x) with respect to x and then evaluate it at x = 5.
First, let's differentiate g(x) using the product rule:
g'(x) = (9x + 7) f'(x) + f(x) * 9
Now, substitute the given values:
g'(5) = (9 * 5 + 7) * 8 + (-6) * 9
Simplifying the expression:
g'(5) = (45 + 7) * 8 - 54
g'(5) = 52 * 8 - 54
g'(5) = 416 - 54
g'(5) = 362
Therefore, g'(5) = 362.