Question

F(x)=x²-1/x+1
F'(x)= Lim h=0 (x+h) -f(x)/h

Answer 1

So the derivative of the function F(x) = x^2 - 1/x + 1 is F'(x) = 2x - (x - 1)/x^2.

Explanation:

The derivative of the function F(x) = x^2 - 1/x + 1 can be found by using the limit definition of the derivative. The derivative is the rate of change of the function at a given point and it is calculated as:

F'(x) = Lim h=0 (x + h)^2 - 1/(x + h) + 1 - (x^2 - 1/x + 1)/h

Expanding the expression inside the limit and simplifying, we get:

F'(x) = Lim h=0 2xh + h^2 - (1 - hx)/(x + h)^2

Using L'Hopital's Rule, we get:

F'(x) = Lim h=0 2x + 2h - (x - 1)/(x + h)^2

As h approaches 0, the limit becomes:

F'(x) = 2x + 2(0) - (x - 1)/x^2 = 2x - (x - 1)/x^2

So the derivative of the function F(x) = x^2 - 1/x + 1 is F'(x) = 2x - (x - 1)/x^2.