47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x) = 1/2x+8; (10,4)

Verified step by step guidance

1. First, understand the relationship between a function and its inverse. If y = f(x), then the inverse function, denoted as f⁻¹(y), satisfies x = f⁻¹(y). The derivative of the inverse function at a point can be found using the formula: (f⁻¹)'(b) = 1 / f'(a), where f(a) = b.
2. Identify the given point (10, 4) in the context of the function f(x) = 1/(2x) + 8. Here, f(a) = b implies f(a) = 4, and a = 10.
3. To find the derivative of the inverse function at the point, first calculate the derivative of the original function f(x). The derivative f'(x) is found using the power rule and the constant rule.
4. Differentiate f(x) = 1/(2x) + 8. The derivative f'(x) = -1/(2x²) because the derivative of 1/(2x) is -1/(2x²) and the derivative of a constant (8) is 0.
5. Evaluate f'(x) at x = 10 to find f'(10). Substitute x = 10 into f'(x) = -1/(2x²) to get f'(10). Use this value to find the derivative of the inverse function at the point (10, 4) using the formula: (f⁻¹)'(4) = 1 / f'(10).

Key Concepts

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x and produces an output y, the inverse function f⁻¹ takes y back to x. Understanding how to find and evaluate inverse functions is crucial for solving problems involving derivatives of inverses.

Derivative of Inverse Functions

The derivative of an inverse function can be calculated using the formula (f⁻¹)'(y) = 1 / f'(x), where y = f(x). This relationship highlights how the rate of change of the inverse function at a point is the reciprocal of the rate of change of the original function at the corresponding point.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is often applied when dealing with inverse functions, as it helps in understanding how changes in one variable affect another.