Final answer:
To find f'(1), we need to take the derivative of the function f(x) = ln(x) x¹⁰ using the product rule and the chain rule. The derivative of ln(x) is 1/x and the derivative of x¹⁰ is 10x⁹. By applying the product rule and substituting x = 1, we find that f'(1) = 1.
Step-by-step explanation:
To find f'(1), we need to take the derivative of the function f(x) = ln(x) x¹⁰. To do this, we can apply the product rule and the chain rule. Let's start by breaking down the function into two parts: f(x) = ln(x) and g(x) = x¹⁰.
The derivative of f(x) = ln(x) is f'(x) = 1/x, and the derivative of g(x) = x¹⁰ is g'(x) = 10x⁹. Now, we can apply the product rule: f'(x)g(x) + f(x)g'(x).
So, f'(x) = (1/x)x¹⁰ + ln(x)(10x⁹). Now, we can substitute x = 1 to find f'(1). Plugging in x = 1, we get:
f'(1) = (1/1)(1¹⁰) + ln(1)(10(1)⁹) = 1 + ln(1) = 1.