Final answer:
The derivatives for the given functions are found by using the chain rule and understanding that the derivative of a constant is zero. For function (a) g(x) = (f(x))^10, the derivative is 10 * (f(x))^9 * (1/x). For function (b) g(x) = f(10), the derivative is 0, and for function (c) h(x) = f(1/x), the derivative is -1/x^3.
Step-by-step explanation:
The student has asked to find expressions for the derivatives of three given functions given that f'(x) = 1/x for x > 0. We will apply the rules of differentiation to each function.
Function (a) g(x) = (f(x))^10
To find the derivative of g(x) = (f(x))^10, we apply the chain rule which states that the derivative of a composite function h(g(x)) is h'(g(x)) × g'(x). Therefore, the derivative of g(x), denoted as g'(x), is 10 × (f(x))^9 × f'(x). Since f'(x) = 1/x, this becomes g'(x) = 10 × (f(x))^9 × (1/x).
Function (b) g(x) = f(10)
For g(x) = f(10), we note that because there is no variable x present in the expression inside the function f, the derivative of g with respect to x is zero. This is because f(10) is a constant with respect to x, and the derivative of a constant is 0.
Function (c) h(x) = f(1/x)
For h(x) = f(1/x), we apply the chain rule again where the outside function is f, and the inside function is 1/x. Here, h'(x) is the derivative of f with respect to 1/x multiplied by the derivative of 1/x with respect to x. This results in h'(x) = f'(1/x) × (-1/x^2). Given that f'(1/x) = x (since f'(x) = 1/x), we simplify to get h'(x) = -1/x^3.