Final answer:
To find the derivatives of the given functions, we can use the power rule and the chain rule. For f(x) = sqrt(x), the derivative is (1/2)x^(-1/2). Evaluating at x = 9 and x = 1, we get f'(9) = 1/6 and f'(1) = 1/2. For f(x) = sin(x), the derivative is cos(x). Evaluating at x = 2 and x = 3, we get f'(2) = -0.416 and f'(3) = -0.989.
Step-by-step explanation:
To find the derivative of the function f(x) = sqrt(x), we can use the power rule for derivatives. The power rule states that if we have a function of the form f(x) = x^n, then the derivative is given by f'(x) = nx^(n-1). Applying this rule to f(x) = sqrt(x), we have f'(x) = (1/2)x^(-1/2). Evaluating this derivative at x = 9, we substitute x = 9 into the derivative function to get f'(9) = (1/2)(9)^(-1/2) = 1/(2*sqrt(9)) = 1/6. Similarly, for x = 1, we have f'(1) = (1/2)(1)^(-1/2) = 1/2.
For the function f(x) = sin(x), the derivative can be found using the chain rule. The chain rule states that if we have a function g(x) = f(h(x)), then the derivative of g(x) is given by g'(x) = f'(h(x)) * h'(x). Applying this rule to f(x) = sin(x), we have f'(x) = cos(x). Evaluating this derivative at x = 2, we have f'(2) = cos(2) = -0.416. For x = 3, we have f'(3) = cos(3) = -0.989.