Final answer:
The question asks for the derivative of the inverse function (f−1)′ at x=5 for the function f(x) = 3 + x + x5. To find this, one would first find the derivative of f(x), f′(x) = 1 + 5x4. Then, we need the value of x such that f(x)=5 to substitute into the formula (f−1)′(y) = 1 / f′(f−1(y)), which we cannot find without solving a complex polynomial equation.
Step-by-step explanation:
The student is asking to find the derivative of the inverse function of f(x) evaluated at x=5, where f(x) is a one to one function defined as f(x) = 3 + x + x5. To find (f−1)′(5), first, we need to find the derivative f′(x) of the original function f(x). Once we have f′(x), we apply the formula for the derivative of the inverse function which is (f−1)′(y) = 1 / f′(f−1(y)). This means we need to determine the value of x such that f(x) = 5 and then substitute this value into the formula.
To solve for f′(x), we differentiate f(x):
f′(x) = d/dx (3 + x + x5)
f′(x) = 1 + 5x4.
Next, we would need to solve for a value of x where f(x) = 5, but since that is not trivial and requires solving a fifth-degree polynomial equation, and we are also not given the additional information to find this value, we cannot proceed further without making assumptions or approximations.