Given that f(x)=2x+cos(x) is one-to-one, use the formula
f(^−1)′(x)=1/ f′(f^−1(x))
find f(^−1)′(x)
To find f(^−1)′(x) using the formula provided, we first need to find f′(x) and f^−1(x).
Step 1: Find f′(x)
To find f′(x), we need to differentiate the function f(x)=2x+cos(x) with respect to x. The derivative of 2x is simply 2, and the derivative of cos(x) is -sin(x). So, the derivative of f(x) is:
f′(x) = 2 - sin(x)
Step 2: Find f^−1(x)
To find f^−1(x), we need to find the inverse function of f(x)=2x+cos(x). The first step is to rewrite the equation in terms of y:
y = 2x + cos(x)
Next, we swap x and y:
x = 2y + cos(y)
Now, solve this equation for y. This process can be quite challenging, and in some cases, it may not be possible to find an explicit equation for the inverse function. However, assuming that an inverse function exists and is expressible, we'll proceed with isolating y:
x - cos(y) = 2y
x = 2y + cos(y)
x - cos(y) = 2y
x = (2y + cos(y))
Unfortunately, solving this equation for y algebraically is not straightforward. So, let's use numerical methods or software to approximate the value of y, denoted as f^−1(x).
Step 3: Use the formula f(^−1)′(x) = 1 / f′(f^−1(x))
Now that we have f′(x) = 2 - sin(x) and f^−1(x), we can plug them into the formula:
f(^−1)′(x) = 1 / (2 - sin(f^−1(x)))
Remember that f^−1(x) represents the value we obtained numerically or using software to approximate the inverse function. Substitute this value in the formula to find f(^−1)′(x).