Sure, let's calculate the first, second, and third derivatives of the function f(x) = 3x - 5/(x - 3).
1. First, let's calculate the first derivative (f ′(x)):
To differentiate 3x with respect to x, we get 3.
And to differentiate -5/(x - 3) with respect to x, the result is 5/(x - 3)^2.
Collectively, the first derivative f ′(x) = 3 + 5/(x - 3)^2.
2. Now let's move on to the second derivative (f ′′(x)):
Differentiating 3 with respect to x, we get 0.
Also, differentiating 5/(x - 3)^2 with respect to x, we get -10/(x - 3)^3.
So the second derivative f ′′(x) = -10/(x - 3)^3.
3. Finally, let's calculate the third derivative (f ′′′(x)):
Differentiating -10/(x - 3)^3 with respect to x, we get 30/(x - 3)^4.
So the third derivative f ′′′(x) = 30/(x - 3)^4.
We have now found the first three derivatives of the given function.
The first derivative f ′(x) = 3 + 5/(x - 3)^2,
The second derivative f ′′(x) = -10/(x - 3)^3,
The third derivative f ′′′(x) = 30/(x - 3)^4.