What is the value of ddx[f−1(x)] when x=2π, given that f(x)=2x−sinx and f−1(2π)=π ?
I got 1/3 is that right?
If g(x) is f^-1(x), and if f(a) = b, then
g'(b) = 1/f'(a)
so, yes, you are correct
To find the value of ddx[f−1(x)] when x=2π, we first need to understand what this expression represents.
The notation ddx represents the derivative of a function with respect to x, which measures the rate at which the function is changing at a specific point. In other words, it calculates the slope of the function's tangent line at that point.
Now, f(x) represents a function, and f−1(x) represents its inverse function. The value f−1(2π)=π tells us that the inverse function of f(x) evaluated at x=2π is equal to π.
To find the value of ddx[f−1(x)] when x=2π, we need to differentiate the inverse function f−1(x) with respect to x and then evaluate it at x=2π.
Since we know that f−1(2π)=π, we can rewrite the expression as ddx[f−1(x)]|x=π.
Now, let's differentiate f(x) = 2x − sin(x) to find the derivative of the original function. The derivative of 2x is 2, and the derivative of sin(x) is cos(x). Therefore, the derivative of f(x) is 2 - cos(x).
To find the derivative of the inverse function, we can use the inverse function theorem, which states that if f(x) and f−1(x) are inverses, then the derivative of f−1(x) is equal to 1 divided by the derivative of f(x) evaluated at f−1(x).
In this case, we want to find ddx[f−1(x)]|x=π, so we need to evaluate the derivative of f(x) at f−1(π). Plugging in π into f(x) gives us f(π) = 2π - sin(π) = 2π.
Now, we can evaluate the derivative of f(x) at x=2π, which is equal to the derivative of 2 - cos(x) evaluated at x=2π. The derivative of 2 is 0, and the derivative of cos(x) is -sin(x). Therefore, the derivative of f(x) at x=2π is 0 - (-sin(2π)) = sin(2π) = 0.
According to the inverse function theorem, we need to take the reciprocal of the derivative of f(x) evaluated at f−1(π). Since the derivative of f(x) at x=2π is 0, the reciprocal is undefined.
Therefore, the value of ddx[f−1(x)] when x=2π is undefined. Your answer of 1/3 is not correct based on the given input.