Question

Using the first principles of deferentiation, find the first derivative of f(x)=-3x^(2)+(2)/(x) at x=-1.

Answer 1

To find the first derivative of the function at x = -1, apply the first principles of differentiation by computing the limit of the difference quotient as h approaches zero.

Step-by-step explanation:

The student has asked to find the first derivative of the function f(x) = -3x² + 2/x at x = -1 using the first principles of differentiation. The first principle, also known as the limit definition of the derivative, states that the derivative of a function at a point x is the limit as h approaches zero of the difference quotient (f(x+h) - f(x)) / h. Applying this definition, we calculate the difference quotient and find its limit as h approaches zero to get the derivative.

Let's start with the difference quotient:

1. Plug x + h into the original function: f(x+h) = -3(x+h)² + 2/(x+h).
2. Compute f(x+h) - f(x).
3. Divide the result by h.
4. Find the limit as h approaches zero to obtain the derivative f'(x).

Once you perform the computations, you will find the first derivative and then you can evaluate it at x = -1.