Find the derivative of the inverse for the function f and real number a.
f(x)=xrad(x-3) a=4
Please help. Thank you very much for your time and assistance.
To find the derivative of the inverse function, we can follow these steps:
Step 1: Start with the given function f(x).
f(x) = x * sqrt(x - 3)
Step 2: First, let's find the inverse function of f(x). To do this, we need to interchange x and y and then solve for y.
x = y * sqrt(y - 3)
Step 3: Solve the equation for y.
x^2 = y^2 * (y - 3)
x^2 = y^3 - 3y^2
y^3 - 3y^2 - x^2 = 0
Step 4: Now, differentiate the equation with respect to x implicitly.
3y^2 * (dy/dx) - 6y * (dy/dx) - 2x = 0
Step 5: Rearrange the equation to isolate (dy/dx).
(dy/dx) * (3y^2 - 6y) = 2x
(dy/dx) = 2x / (3y^2 - 6y)
Step 6: Plug in the value of a into the equation and evaluate.
(dy/dx) = 2(4) / (3(4)^2 - 6(4))
(dy/dx) = 8 / (48 - 24)
(dy/dx) = 8 / 24
(dy/dx) = 1/3
So, the derivative of the inverse function of f(x) when x = 4 is 1/3.
Note: This is the derivative of the inverse function, not the original function.