Final answer:
To find the derivative of an inverse function at a certain point, use the reciprocal of the derivative of the original function at the corresponding point, using the formula d/dx(f⁻¹(y)) = 1/(f′(f⁻¹(y))). This leverages the chain rule of calculus for efficiency and avoids the need for finding the full inverse function.
Step-by-step explanation:
To find the derivative of an inverse function at a particular point without explicitly finding the inverse function, you can use the formula d/dx(f⁻¹(y)) = 1/(f′(f⁻¹(y))). In other words, the derivative of the inverse function at a point is the reciprocal of the derivative of the function at the corresponding point of the original function. This is based on the chain rule of calculus which asserts that the derivative of a function and its inverse function, when multiplied together, equals 1.
For example, if you have a function f(x) and you know the value of its derivative f′(x) at x = a, and you also know that f(a) = b, then the derivative of the inverse function f⁻¹ at b is 1/f′(a). No need to explicitly find the entire inverse function; only the original function's derivative at a certain point is required.
This method is often used in calculus problems where the function's inverse would be too complicated to find or unnecessary for the task at hand. It allows for the computation of instantaneous rates of change or slopes of tangent lines without having to derive and solve for the inverse function fully.