Find the value of df^-1/dx at x=f(a).
f(x)=5x+8, a=-1
i got 5?
To find the value of df^-1/dx at x=f(a), we need to differentiate the inverse function of f(x) with respect to x, and then evaluate it at x=f(a).
Step 1: Find the inverse function of f(x).
The inverse function of f(x) can be found by swapping x and y and solving for the new y.
So, let's start by swapping x and y in the equation f(x) = 5x + 8:
x = 5y + 8
Next, solve for y:
x - 8 = 5y
y = (x - 8)/5
So, the inverse function of f(x) is f^-1(x) = (x - 8)/5.
Step 2: Differentiate the inverse function.
To differentiate f^-1(x), we can use the power rule for differentiation. The power rule states that if we have a function of the form f(x) = ax^n, then the derivative is given by f'(x) = n * ax^(n-1).
In our case, f^-1(x) = (x - 8)/5 is in the form f(x) = ax^n, where a = 1/5 and n = 1. Thus, we can differentiate f^-1(x) as follows:
f'^-1(x) = (1)(1/5)(x - 8)^(1-1)
f'^-1(x) = 1/5
Step 3: Evaluate f'^-1(x) at x = f(a).
To evaluate f'^-1(x) at x = f(a), we need to plug in the value of f(a) into f'^-1(x).
Given that f(x) = 5x + 8 and a = -1, let's compute f(a):
f(a) = 5(-1) + 8 = -5 + 8 = 3
Now, substitute f(a) = 3 into f'^-1(x):
f'^-1(x) |x=f(a) = 1/5 |x=3 = 1/5
Therefore, the value of df^-1/dx at x = f(a) is 1/5, not 5.