SOLUTION: Given f(x) = x^2 +3, find and simplify the difference quotient ([f(x+h) - f(x)])/h. What is the best way to go about solving this? I know I need to substitute the equations toge

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Question 1202652: Given f(x) = x^2 +3, find and simplify the difference quotient ([f(x+h) - f(x)])/h.
What is the best way to go about solving this? I know I need to substitute the equations together like (x+h)^2 + 3 - (x^2=3)
I don't know if that substitution is correct. Any help with this problem would be greatly appreciated.

Found 2 solutions by Theo, Alan3354:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
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 https://www.derivative-calculator.net/ 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

Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
Given f(x) = x^2 +3, find and simplify the difference quotient ([f(x+h) - f(x)])/h.
------------------
f(x) = x^2 +3
f(x+h) = (x+h)^2 + 3 = x^2 + 2hx + h^2 + 3
Subtract f(x)
---> x^2 + 2hx + h^2 + 3 - (x^2+3) = 2hx + h^2
Divide by h ---> 2x + h