derivative of inverse
The derivative of the inverse of a function can be found using the inverse function theorem
The derivative of the inverse of a function can be found using the inverse function theorem. Let’s say we have a function f(x) and its inverse function is denoted as f^(-1)(x).
The inverse function theorem states that if a function f(x) is differentiable at a point c and f'(c) is not equal to 0, then its inverse function f^(-1)(x) is also differentiable at the corresponding point d = f(c), and the derivative of the inverse function at d is given by:
(f^(-1))'(d) = 1 / f'(c)
So, to find the derivative of the inverse function f^(-1)(x), we need to determine an appropriate value c for which we can evaluate f'(c) and find its inverse f^(-1)(x) at the corresponding d.
Let’s go through an example to illustrate this:
Example:
Consider the function f(x) = 2x + 3. We want to find the derivative of its inverse function.
Step 1: Find f'(x)
To find f'(x), we can simply take the derivative of the function f(x):
f'(x) = 2
Step 2: Find an appropriate value of c
To find an appropriate value of c, let’s choose a value for x. Let’s say x = 2. So, f(x) = f(2) = 2(2) + 3 = 7.
Step 3: Evaluate f'(c) at c = 2
Now, we need to evaluate f'(c) at c = 2:
f'(2) = 2
Step 4: Find the corresponding d
The corresponding d value in our case is the output of the function f^(-1)(x) when x = 7. Therefore, we need to determine the inverse function f^(-1)(x) and find f^(-1)(7).
To find the inverse function, we need to interchange the positions of x and y in the original function, and then solve for y:
x = 2y + 3
x – 3 = 2y
(y – 3/2) = (1/2)x
y = (1/2)x – 3/2
So, the inverse function is f^(-1)(x) = (1/2)x – 3/2.
Substituting x = 7 into the inverse function, we have:
f^(-1)(7) = (1/2)(7) – 3/2 = 7/2 – 3/2 = 4/2 = 2
Therefore, the corresponding d in this case is 2.
Step 5: Calculate the derivative of the inverse
Now that we have f'(c) = f'(2) = 2 and the corresponding d = f^(-1)(7) = 2, we can use the inverse function theorem to find the derivative of the inverse:
(f^(-1))'(2) = 1 / f'(c) = 1 / f'(2) = 1 / 2
So, the derivative of the inverse function f^(-1)(x) at x = 2 is 1/2.
In summary, the derivative of the inverse function of f(x) = 2x + 3 at x = 2 is 1/2.
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