`f(x) = x^2 + x - 3` Find the derivative of the function by the limit process.

Textbook Question

Chapter 2, 2.1 - Problem 17 - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385's profile pic

hkj1385 | (Level 1) Assistant Educator

Posted on

By limit process, the derivative of a function f(x) is :-

f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]

Now, the given function is :-

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

thus, f'(x) = lim h --> 0  [{{(x+h)^2} + (x+h) - 3} - ((x^2) + x - 3)}/h]

or, f'x) = lim h --> 0 [{{(x+h)^2} - (x^2) + h}/h]

or, f'(x) = lim h --> 0 [{(x^2) + (h^2) + 2xh + h - (x^2)}/h]

or,  f'(x) = lim h --> 0 [{(h^2) + h + 2xh}/h] = [h + 1 + 2x]

putting the value of h = 0 in the above  expression we get

f'(x) = 2x + 1

scisser's profile pic

scisser | (Level 3) Honors

Posted on

Although you asked for the limit process, I definitely think there is some benefit in checking your answer using another method. For this, I would use The Power Rule, which states:

if

` y=x^(n)`

then

` y= n*x^(n-1)`

So,

if ` f(x)=x^(2)+x-3`

then,

` f'(x)=2x+1`

Thus, your answer has been confirmed and is correct.

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