Finding the Derivative of an Inverse Function

In summary, the conversation discussed finding the inverse of a function and using that to find the derivative of the inverse function. The two ways to find the derivative of the inverse were discussed, with the recommendation to specify which value is being used when solving for g'(4).
  • #1
UrbanXrisis
1,196
1
I was taught in class that if g(x) is f^-1(x), then g'(y)=1/f'(x)

my notes for an example:
f(x)=x^2
g(x)=f^-1(x)=x^(.5)
g’(4)=?
g(x)= x^(.5)
g’(x)=.5x^-.5
g’(4)=1/f’(x), x=4^(.5)
g’(4)=1/f'(2)=1/2*2=1/4

I have no clue how g(x)=f^-1(x)=x^(.5) and how g’(4)=1/f'(2)

I am just overall confused any help would be appreciated
 
Physics news on Phys.org
  • #2
f(x) = x^2

set f(x) = y

y = x^2

(*To find inverse solve for x explicitly in terms of y*)

y^0.5 = x

f^-1(x) = x^0.5

let g(x) = f^-1(x)

g(x) = x^0.5

The second one follows the same way
 
  • #3
I have:
g’(4)=1/f'(2)=1/2*2=1/4

and also:
g'(4)=1/f'(x)=1/f'(2)=1/4

is both ways acceptable?
 
  • #4
UrbanXrisis said:
I have:
g’(4)=1/f'(2)=1/2*2=1/4

and also:
g'(4)=1/f'(x)=1/f'(2)=1/4

is both ways acceptable?

I wouldn't go with the second one. It could be seen as wrong to say g'(4) = 1/f'(x) without saying what x is explicitly, which in this case is root(4).

But, it depends how picky your instructor is
 
  • #5
what about:
g’(x)=.5x^-.5=.5(4)^-.5=1/4

could that be acceptable too?
 
  • #6
UrbanXrisis said:
what about:
g’(x)=.5x^-.5=.5(4)^-.5=1/4

could that be acceptable too?

Sure,

g(x) = [tex]x^\frac{1}{2}[/tex]

thus,

g'(x) = [tex]\frac{1}{2x^\frac{1}{2}}[/tex]

g'(4) = [tex]\frac{1}{2*4^\frac{1}{2}}[/tex]

g'(4) = [tex]\frac{1}{2*2}[/tex]

g'(4) = [tex]\frac{1}{4}[/tex]


Again, I would just recommend that you specify you are changing from the general equation g'(x) to g'(4) somewhere in your solution. But if its not on a test, or your teacher isn't picky then that should be fine.
 

What is the derivative of an inverse function?

The derivative of an inverse function is equal to the reciprocal of the derivative of the original function, evaluated at the corresponding input of the inverse function.

How do you find the derivative of an inverse function using the chain rule?

To find the derivative of an inverse function using the chain rule, you first rewrite the inverse function in terms of the original function. Then, you apply the chain rule by taking the derivative of the original function and multiplying it by the derivative of the inverse function.

Can the derivative of an inverse function be negative?

Yes, the derivative of an inverse function can be negative. This can happen when the original function has a decreasing slope, causing the inverse function to have a decreasing slope as well.

What is the relationship between the derivative of a function and the derivative of its inverse?

The derivative of an inverse function is the reciprocal of the derivative of the original function. This means that the slopes of the original and inverse functions at corresponding points are reciprocals of each other.

How is the derivative of an inverse function related to the concept of inverse functions?

The derivative of an inverse function is a mathematical tool used to find the rate of change of the inverse function. It is related to the concept of inverse functions because it allows us to find the slope of the inverse function at a specific input value, which is useful in many real-world applications.

Similar threads

  • Introductory Physics Homework Help
Replies
32
Views
958
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
308
  • Introductory Physics Homework Help
Replies
8
Views
778
  • Introductory Physics Homework Help
Replies
6
Views
535
  • Introductory Physics Homework Help
Replies
10
Views
863
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
634
  • Introductory Physics Homework Help
Replies
7
Views
621
  • Introductory Physics Homework Help
Replies
8
Views
874
Back
Top