Final answer:
To find the derivative of the function f(x) = 5 sin(x) ln(2x), we use the product rule. The derivative is 5 cos(x) ln(2x) + 5 sin(x) * (1/(2x) * 2).
Step-by-step explanation:
To find the derivative of the function f(x) = 5 sin(x) ln(2x), we use the product rule. The product rule states that if we have two functions multiplied together, their derivative is the first function times the derivative of the second.
Plus the second function times the derivative of the first. Let's start by finding the derivative of sin(x). The derivative of sin(x) is cos(x). Next, let's find the derivative of ln(2x). The derivative of ln(u) is 1/u times the derivative of u.
In this case, u = 2x, so the derivative of ln(2x) is 1/(2x) times the derivative of 2x, which is just 2. Now, using the product rule, we can find the derivative of f(x) = 5 sin(x) ln(2x). It is: f'(x) = 5 cos(x) ln(2x) + 5 sin(x) * (1/(2x) * 2).