Final answer:
To find the value of C for which F'(4π) = 9, we need to find the derivative of F(x) and then solve the equation for C. We can find the derivative by adding the derivatives of Cx and Ln(Cos(x)) together, and then substitute x = 4π into the equation to solve for C.
Step-by-step explanation:
To find the value of C for which F'(4π) = 9, we need to first find the derivative of F(x). The derivative of Cx is just C, and the derivative of Ln(Cos(x)) can be found using the chain rule. The derivative of Ln(u) is 1/u, and the derivative of Cos(x) is -Sin(x). So, the derivative of Ln(Cos(x)) is -Sin(x)/Cos(x), which simplifies to -Tan(x).
Now, we can find the derivative of F(x) by adding the derivatives of Cx and Ln(Cos(x)) together. So, F'(x) = C - Tan(x).
To find the value of C for which F'(4π) = 9, we need to substitute x = 4π into the derivative equation and solve for C. So, we have 9 = C - Tan(4π). Solving this equation will give us the value of C that satisfies the condition.