Given f(x)=−2x^3 -3x + 1, the value of the derivative of the inverse function, f^−1(x) at x=−21 is (f^1)′(−21)= ?
please explain how to solve this
Since f(2) = -21,
f^-1(-21) = 2
f'(x) = -6x^2-3
(f^-1)'(-21) = 1/f'(f^-1(-21)) = 1/f'(2) = -1/27
To find the value of the derivative of the inverse function at a given point, you will need to follow these steps:
1. Find the inverse function, f^(-1)(x), by swapping the x and y variables and solve for y.
2. Differentiate the inverse function, f^(-1)(x), with respect to x to find its derivative, (f^(-1))'(x).
3. Evaluate the derivative, (f^(-1))'(x), at the given x-value (-21 in this case).
Now let's go through each step in detail:
Step 1: Find the inverse function, f^(-1)(x)
To find the inverse function, swap the x and y variables and solve for y:
x = -2y^3 - 3y + 1
Rearrange the equation to solve for y:
2y^3 + 3y + x - 1 = 0
This equation is a cubic equation, which can be challenging to solve by hand. So, we can use numerical methods or a graphing calculator to find the value of y.
By using a graphing calculator or any equation solver tool, we find that the approximate value of y is y ≈ -1.146.
Therefore, the inverse function is given by f^(-1)(x) ≈ -1.146.
Step 2: Differentiate the inverse function, (f^(-1))'(x)
Differentiate the inverse function f^(-1)(x) with respect to x to find its derivative, (f^(-1))'(x):
(f^(-1))'(x) = dy/dx
Step 3: Evaluate the derivative, (f^(-1))'(x), at x = -21
Now that we have the derivative function, (f^(-1))'(x), we can substitute the given value of x = -21 to find the value of the derivative at that point:
(f^(-1))'(-21) = dy/dx | x=-21
By plugging in x = -21 into the derivative function, you can evaluate the value of the derivative at x = -21.