Given y = 5x +25, find its inverse function then find the derivative.
To find the inverse function, we need to switch the x and y variables and solve for y.
So, the original equation is:
y = 5x + 25
Switching variables, we get:
x = 5y + 25
Now, solve for y:
x = 5y + 25
x - 25 = 5y
y = (x - 25) / 5
Therefore, the inverse function is:
f^(-1)(x) = (x - 25) / 5
To find the derivative of this inverse function, we can use the formula for the derivative of an inverse function:
Let y = f(x) and let g = f^(-1)(x), then
g'(x) = 1 / f'(g(x))
The derivative of the original function y = 5x + 25 is 5. Therefore, the derivative of the inverse function f^(-1)(x) = (x - 25) / 5 is:
g'(x) = 1 / 5
So, the derivative of the inverse function is:
f^(-1)(x) = 1 / 5