For the function f(x) = sqrt(9-x), which is equal to d/dx(f-1( -1 is the inverse)(x))
a) -2x
b) sqrt(x) + 9
c) x^2 + 9
d) 2x
e) 9-x^2
Yep I was correct
well lets see. Since f^-1(x) = 9-x^2 then the derivative of 9-x^2 is -2x correct?
-2x
The inverse function is x = sqrt(9-y)
If you differentiate the inverse function, you would get
y' = -2 * sqrt(9-y)
Then substitue x = sqrt(9-y)
y' = -2x
Hey there! Let's figure out the derivative of the inverse of f(x) = sqrt(9-x), which is f^(-1)(x).
To find the derivative, we can use the formula: d/dx(f^(-1)(x)) = 1 / (d/dy(f(y))), where y = f^(-1)(x).
In this case, let's find the value of y first. So, we have y = f^(-1)(x), which gives us x = sqrt(9-y). Solving this equation for y gives us y = 9 - x^2.
So, the function f^(-1)(x) is equal to 9 - x^2.
Now, let's find the derivative of f^(-1)(x) = 9 - x^2:
d/dx(9 - x^2) = -2x.
Therefore, the correct answer is (a) -2x.
Hope that puts a smile on your face! If you have any more questions, feel free to ask!
To find the derivative of the inverse function, we can use the formula:
(d/dx)(f^(-1)(x)) = 1 / (dy/dx)
First, let's find the derivative of f(x) = sqrt(9-x):
f(x) = sqrt(9-x)
To find the derivative, we can use the chain rule:
Let u = 9-x.
Then, f(u) = sqrt(u), and f(x) = f(u(x)).
Using the chain rule, the derivative of f(x) is:
f'(x) = d(sqrt(u))/du * du/dx
The derivative of sqrt(u) with respect to u is:
d(sqrt(u))/du = 1 / (2*sqrt(u))
The derivative of u = 9-x is:
du/dx = -1
Now, we can substitute these derivatives back into the chain rule formula:
f'(x) = (1 / (2*sqrt(u))) * (-1)
Since u = 9-x, we can rewrite the expression as:
f'(x) = (1 / (2*sqrt(9-x))) * (-1)
Next, to find the derivative of the inverse function, we flip the roles of x and f(x) and solve for the derivative:
d/dx(f^(-1)(x)) = 1 / (df/dx)
Therefore,
d/dx(f^(-1)(x)) = 1 / (1 / (2*sqrt(9-x))) * (-1)
Simplifying further, we get:
d/dx(f^(-1)(x)) = -2*sqrt(9-x)
So, the answer is:
d/dx(f^(-1)(x)) = -2*sqrt(9-x)
Therefore, the correct option is a) -2x.