You should just take the derivative of both sides. Use the chain rule on the left hand side... Where did your equality go, and where did the -x come from?harrietstowe said:Homework Statement
f is a function with an inverse and it is differentiable. Use f(f-1(x))=x
and come up with the formula for the derivative of f-1
Homework Equations
The Attempt at a Solution
I tried expanding that equation to f'(f-1(x))*f'-1(x) -x
but I tested this and it didn't work.
Thanks
An inverse function in calculus is a function that “undoes” another function. In other words, if the original function takes an input and produces an output, the inverse function does the opposite, taking the output as its input and producing the original input.
To find the inverse of a function, you can follow these steps:
1. Write the function as y = f(x).
2. Swap the x and y variables.
3. Solve the new equation for y.
4. The resulting equation is the inverse function.
Note: Not all functions have inverses, so it is important to check for one-to-one correspondence before finding the inverse.
The derivative of an inverse function can be found using the formula:
(f^-1)'(x) = 1 / f'(f^-1(x))
In words, this means that the derivative of an inverse function at a given point is equal to the reciprocal of the derivative of the original function at the corresponding point on the inverse function.
To solve calculus problems with inverse functions, you can use the following steps:
1. Find the inverse function.
2. Use the formula for the derivative of an inverse function to find the derivative.
3. Substitute the given input value into the derivative to find the slope at that point.
4. Use the point-slope formula to find the equation of the tangent line at that point.
5. Use the equation of the tangent line to solve the problem.
Real-world applications of calculus inverse function derivative problems include optimization, physics, economics, and engineering. For example, in economics, inverse functions and their derivatives are used to determine the maximum profit for a business or the optimal production level. In physics, they are used to find the rate of change of a quantity with respect to another quantity, such as velocity with respect to time. In engineering, they are used to analyze the behavior of systems and design optimal solutions.