Derivative of the inverse of a function

In summary, we discussed the function y=x+(1/x) and its inverse, y^-1, at the point y=17/4. After finding the derivative of y, we substituted the given value of y to calculate an approximate value for the derivative. However, in attempting to find the inverse function, we realized that the function did not have a unique inverse and would require further solving to find the correct inverse.
  • #1
yurkler
2
0

Homework Statement


y=x+(1/x) at y=17/4

Homework Equations



The Attempt at a Solution


y^-1: x=y+(1/y)

differentiate: 1=y'+ln(y)y'
1=y'(1+ln(y))
y'=1/(1+ln(y))

put that over 1: 1+ln(y)

plug in y: 1+ln(17/4)
=approximately 2.447
 
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  • #2
yurkler said:
the function is y=x+(1/x) at y=17/4

attempt:
y^-1: x=y+(1/y)
Here, by switching letters, you have x = f(y). f is the same function as above, except that its argument is now called y.
yurkler said:
differentiate: 1=y'+ln(y)y'
You differentiated with respect to x, so what you have above is 1 = f'(y)*dy/dx
yurkler said:
1=y'(1+ln(y))
y'=1/(1+ln(y))
Now you have dy/dx = 1/f'(y)
yurkler said:
put that over 1: 1+ln(y)
What you're saying and what you're doing are two different things. If you put anything over 1, you get exactly the same thing.

What you did is take the reciprocal. Why?
yurkler said:
plug in y: 1+ln(17/4)
=approximately 2.447

You switched letters, so what was the old y value (17/4) is now an x value. The question is, what is the new y value?

The answer I get is ~.42
 
  • #3
The inverse function is NOT given by x = y + 1/y

To see this, just use a test value. e.g., if x = 2, then y(2) will give you some value. If you then plug this value into your "inverse" function (i.e. x(y(2)), you should get back 2. But you don't, because:

y(2) = 2 + 1/2 = 5/2

x(5/2) = 5/2 + 2/5 = 25/10 + 4/10 = 29/10

29/10 is not equal to 2. Since you didn't get back what you started with, your "inverse" function x(y) must be wrong.

To get the actual inverse function, solve the equation to find x in terms of y. In other words, solve the equation for x. Hint: you'll find that the function does not have a unique inverse.
 
  • #4
Mark44 said:
The answer I get is ~.42

42!
 

Related to Derivative of the inverse of a function

1. What is the derivative of the inverse of a function?

The derivative of the inverse of a function is the reciprocal of the derivative of the original function. In other words, if the original function is f(x), then the derivative of the inverse function is 1/f'(x).

2. How do you find the derivative of the inverse of a function?

To find the derivative of the inverse of a function, you can use the following formula: (f^-1)'(x) = 1/f'(f^-1(x)). This means that you first find the derivative of the original function, substitute the inverse function in for x, and then take the reciprocal of that value.

3. Why is it important to know the derivative of the inverse of a function?

Knowing the derivative of the inverse of a function is important because it allows you to find the slope of the tangent line to a curve at a specific point. This is useful in many areas of mathematics and science, such as optimization and curve fitting.

4. Are there any special rules for finding the derivative of the inverse of a function?

Yes, there are a few special rules for finding the derivative of the inverse of a function. For example, if the original function is a polynomial, then the derivative of the inverse function is also a polynomial. Additionally, the chain rule must be used when finding the derivative of the inverse of a composite function.

5. Can the derivative of the inverse of a function be negative?

Yes, the derivative of the inverse of a function can be negative. This would mean that the slope of the tangent line to the inverse function is negative at a particular point. However, it is important to note that the derivative of the inverse of a function is always less than or equal to zero, as the original function must be a one-to-one function for it to have an inverse.

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