Question 1202652
i believe it works like this.


you want the limit of (f(x + h) - f(x)) / h as h approaches 0.


your equation if f(x) = x^2 + 3


f(x+h) is derived by replacing x with (x + h) in the original equation to get:


f(x+h) = (x+h)^2 + 3


simplify this to get f(x+h) = x^2 + 2hx + h^2 + 3


(f(x + h) - f(x)) / h becomes (x^2 + 2hx + h^2 + 3 - (x^2 + 3)) / h which becomes:


(x^2 + 2hx + h^2 + 3 - x^2 - 3) / h


combine like terms to get:


2hx + h^2 / h


h from the numerator and the denominator cancel out and you are left with:


2x + h


as h approaches 0, this becomes 2x.


that's your derivative.


the results from the derivative calculator at <a href = "https://www.derivative-calculator.net/" target = "_blank">https://www.derivative-calculator.net/</a> confirm that the derivative is 2x.


the derivative is the same as the limit of [f(x+h) - f(x)] / h, as h approaches 0.


let me know if you still have any questions regarding this.


theo