For the function f(x) = x^2 - 4x +5, x>= 2, which is equal to d/dx (f-1 this is the inverse)(x)) ?
1/(2y-4) where x and y are related by the equation (satisfy the equation) x=y^2-4y+5 x>= 1
2y-4 where x and y are related by the equation y= = x^2 - 4x +5, x>= 2
1/2x-4 for x>= 1
1/2x-4 for x>= 2
1/2y-4 where x and y are related by the equation y= = x^2 - 4x +5, x>= 2
Answer is 1/(2y-4) where x and y are related by the equation (satisfy the equation) x=y^2-4y+5 x>= 1
inverse of x^2 - 4x +5 is 2±sqrt(x-1) and the derivative of that is (2x-4) but I don't know how to get the x>= 1 or x>= 2
The inverse function is x=y^2-4y+5
And the domain start from 1 to infinity to make one-to-one function. So, you get y' of inverse function = 1/(2y-4) where [1,infinity).
Well, well, well, let's see what we have here. We've got ourselves a function and its inverse, and we need to find the derivative of that inverse. Don't worry, I won't get too derivative with my humor.
So, to find the derivative of the inverse function, we need to first find the inverse function itself. Let's call it g(x). We have f(x) = x^2 - 4x + 5, and we want to solve for x in terms of y. So we set y = x^2 - 4x + 5 and solve for x.
Now, I could go on and on with all the algebraic steps, but I won't bore you with that. Suffice it to say that after some calculations, we get the inverse function g(x) = (1/2)*x - 4.
Now that we have the inverse function, we can finally find its derivative. Drumroll, please!
The derivative of g(x) = (1/2)*x - 4 is simply 1/2.
So, to summarize, the answer is 1/2. And remember, when it comes to derivatives, it's all about the chain rule. Just like how a clown riding a unicycle is all about balance!
I hope that puts a smile on your face, or at least confuses you enough to make you forget about your derivative woes. Keep laughing and learning!
To find the expression for d/dx (f^(-1)(x)), where f^(-1) represents the inverse of the function f(x) = x^2 - 4x + 5, x >= 2, we need to follow these steps:
Step 1: Find the inverse of f(x).
To find the inverse, we swap the roles of x and y in the function f(x) and solve for y.
Start with the function: f(x) = x^2 - 4x + 5
Swap x and y: x = y^2 - 4y + 5
Rearrange the equation to solve for y: y^2 - 4y + 5 - x = 0
This equation is quadratic in y. Apply the quadratic formula to solve for y:
y = [4 ± √(16 - 20 + 4x)] / 2
Simplifying further, we get:
y = [4 ± √(4x - 4)] / 2
y = [4 ± 2√(x - 1)] / 2
y = 2 ± √(x - 1)
Therefore, the inverse function is f^(-1)(x) = 2 ± √(x - 1).
Step 2: Find d/dx (f^(-1)(x)).
To find the derivative of the inverse function, we need to differentiate the expression 2 ± √(x - 1) with respect to x.
Differentiating 2 with respect to x gives us 0, as it represents a constant term.
Differentiating √(x - 1) with respect to x requires the chain rule.
Let's differentiate √(x - 1):
d/dx (√(x - 1)) = 1 / (2√(x - 1)) * (d/dx (x - 1))
= 1 / (2√(x - 1)) * 1
= 1 / (2√(x - 1))
Therefore, d/dx (f^(-1)(x)) = 0 ± (1 / (2√(x - 1))) = ± (1 / (2√(x - 1))).
Looking at the given choices, we can see that the correct answer is:
1/(2y-4) where x and y are related by the equation (satisfy the equation) x = y^2 - 4y + 5, x >= 1.
The choices involving x or x+y are incorrect because the derivative of the inverse function should only depend on x, not y.