Finding the derivative of an unknowable inverse function

In summary, the conversation discusses finding the inverse function of a given function and using the derivative rule for inverses to find the value of g'(5). It is suggested to try simple values of x to find an x where f(x) equals 5, which can then be used to solve for g'(5). The final solution is g'(5) = 1/11.
  • #1
yiyopr
4
0

Homework Statement



The function f(x) has an inverse function, g(x). Find g'(5).

Homework Equations



[itex]f(x) = x^5 + 2x^2 + 2x[/itex]

The Attempt at a Solution



I don't see how I can possibly find the inverse of this function. So I opted to use the derivative rule for inverses.

[itex] f'(x) = 5x^4 + 4x + 2[/itex][itex] 5 = x^5 + 2x^2 + 2x[/itex]This doesn't help me either. I need to solve for x in the second equation and substitute that x in the derivative. In essence, I can't find x of f(x), and without that I can't find the value of f'(x), which is the reciprocal of g'(x).

Any help would be appreciated!
 
Last edited:
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  • #2
Try a few simple values of x, you will quickly find an x where f(x)=5.
 
  • #3
LCKurtz said:
Try a few simple values of x, you will quickly find an x where f(x)=5.

Oh wow. x=1. That was very foolish of me! Thank you for your time man.
 
  • #4
The answer is g'(5)=1/11 . If anyone is curious.
 

FAQ: Finding the derivative of an unknowable inverse function

What is the purpose of finding the derivative of an unknowable inverse function?

The purpose of finding the derivative of an unknowable inverse function is to determine the rate of change of the original function at a specific point. This can be useful in many applications, such as optimization problems and curve fitting.

How is the derivative of an unknowable inverse function calculated?

The derivative of an unknowable inverse function can be calculated using the inverse function rule, which states that the derivative of an inverse function is equal to the reciprocal of the derivative of the original function evaluated at the corresponding point.

Can the derivative of an unknowable inverse function be calculated for all types of functions?

No, the derivative of an unknowable inverse function can only be calculated for functions that are invertible, meaning they have a one-to-one mapping between their input and output. Functions that are not invertible, such as horizontal lines, do not have a well-defined inverse and therefore do not have a derivative.

Is there a general formula for finding the derivative of an unknowable inverse function?

No, there is not a general formula for finding the derivative of an unknowable inverse function. The method used to find the derivative will depend on the specific function and the techniques used to find its inverse.

What are some common techniques for finding the derivative of an unknowable inverse function?

Some common techniques for finding the derivative of an unknowable inverse function include the use of the inverse function rule, implicit differentiation, and the substitution method. It is also helpful to have a good understanding of algebra and trigonometry to simplify the expressions involved.

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