Calc II Question: Inverse Functions and Derivatives Explained | Help Needed

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In summary, the question deals with inverse functions and finding the derivative of G(x) = 1/g(x) when given f(x) = 3 and f'(x) = 1/9 at x = 3. The formula g'(x) = 1/f'(g(x)) can be used, but the presence of G(x) is causing confusion. Using the chain rule, it can be derived that G'(x) = -1/g(x)^2 * g'(x). Substituting g(x) = f^-1(x), g'(x) = 1/f'(x), the final answer is G'(3) = -1/(3^2 * 1/9) = -1
  • #1
frasifrasi
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This deals with inverse functions:

suppose g(x) is the inverse of f(X) and G(X) = 1/g(X). If f(3) = 3 and f'(3) = 1/9, find G'(3).

Does anyone know how to answer this question?

Thanks.

I was thinking of using the formula g'(x) = 1/f'(g(X)), but the G(X) is throwing me off.
 
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  • #2
[tex]f'(x)= \frac{1}{f^{-1}'(x)}[/tex]
 
  • #3
tthat is not what is being asked...
 
  • #4
frasifrasi said:
This deals with inverse functions:

suppose g(x) is the inverse of f(X) and G(X) = 1/g(X). If f(3) = 3 and f'(3) = 1/9, find G'(3).

Does anyone know how to answer this question?

Thanks.

I was thinking of using the formula g'(x) = 1/f'(g(X)), but the G(X) is throwing me off.

If G(x)= 1/ g(x)= g-1 then, by the chain rule, G'= -1g(x)-2 g'(x). Since g(x) is f-1(x), g'(x)= 1/f'(x).
 
  • #5
I still don't get it. Can anyone explain it using the actual numbers to derive an answer? I need to understand this before the exam.
 

FAQ: Calc II Question: Inverse Functions and Derivatives Explained | Help Needed

What are inverse functions?

Inverse functions are functions that "undo" each other. In other words, if you have a function f(x), its inverse function is denoted as f^-1(x) and it essentially reverses the operation of f(x). For example, if f(x) = 2x, then f^-1(x) = x/2.

How are inverse functions related to derivatives?

Inverse functions are closely related to derivatives because the derivative of an inverse function can be found by taking the reciprocal of the derivative of the original function. In other words, if f(x) and f^-1(x) are inverse functions, then (f^-1)'(x) = 1/f'(x).

How do you find the derivative of an inverse function?

To find the derivative of an inverse function, you can use the formula (f^-1)'(x) = 1/f'(x). This means you first find the derivative of the original function, then take its reciprocal to find the derivative of the inverse function. In some cases, you may also need to use the chain rule.

Can you give an example of finding the derivative of an inverse function?

Sure! Let's say we have the function f(x) = 3x^2. To find the derivative of its inverse function, we first find the derivative of f(x), which is f'(x) = 6x. Then, we take the reciprocal to find the derivative of the inverse function: (f^-1)'(x) = 1/6x.

What is the importance of understanding inverse functions and derivatives?

Understanding inverse functions and derivatives is crucial in many areas of mathematics and science, especially in calculus and related fields. These concepts are used to solve equations, find maximum and minimum values, and understand the behavior of functions. They are also important in applications such as physics and engineering, where they are used to model and analyze real-world phenomena.

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