- #1
Shannabel
- 74
- 0
Homework Statement
let f(x)=(4t^3+4t)dt(between 2 and x)
if g(x) = f^(-1)(x), then g'(0)=?
Homework Equations
The Attempt at a Solution
f'(x) = 4x^3+4x
annd i already don't know where to go from here.. help?
Bohrok said:There's a formula for the derivative of inverses
http://en.wikipedia.org/wiki/Inverse_functions_and_differentiation"
If you start with f(f-1(x)) = x, differentiate both sides and rearrange and you'll get something like
Shannabel said:so
[f^(-1)(0)]' = 1/[f'(f^(-1)(0))]
but where do i go from here?
because i don't know what f^(-1)(0) is...
Bohrok said:If f(b) = 0, then taking the inverse of both sides gives you f-1(0) = b. Then you apply this to the original function you were given to find f-1(0)
Shannabel said:at the beginning you started with f(f^(-1)(x))=x
... where did that come from?
Bohrok said:That's the purpose of the inverse functions: the compositions of inverse functions return the input x, f(f-1(x)) = f-1(f(x)) = x. http://en.wikipedia.org/wiki/Inverse_function" has a pretty good article with examples.
Integration with inverse functions is the process of finding the original function when given its derivative. It involves using inverse functions to solve for the original function and then integrating it to find the general solution.
Integration with inverse functions involves finding the inverse of the derivative function, while regular integration involves finding the anti-derivative of a function. This means that in integration with inverse functions, we are finding the original function, while in regular integration, we are finding a function whose derivative is the given function.
Integration with inverse functions is important because it allows us to solve for the original function when given its derivative. This is useful in many applications, such as in physics, engineering, and economics, where we often have information about the derivative of a function but need to find the original function.
The process for integrating with inverse functions involves using inverse functions to solve for the original function and then integrating it to find the general solution. This usually involves using algebraic manipulation and inverse function rules to solve for the original function.
Yes, there are special techniques for integrating with inverse functions, such as substitution and integration by parts. These techniques can make the process of solving for the original function easier and more efficient.