"Using the limit definition, how do you find..." was answered by our community

Dashawn Robbins

Answered question

2022-04-22

Using the limit definition, how do you find the derivative of

y = x^{2} + x + 1

Answer & Explanation

eldgamliru9x

Beginner2022-04-23Added 16 answers

Explanation:
Find

f (a + h) = {(a + h)}^{2} + (a + h) + 1 = a^{2} + 2 a h + h^{2} + a + h + 1

and

f (a) = a^{2} + a + 1

then plug it in to the formula

lim_{h \to 0} \frac{f (a + h) - f (a)}{h}

and then simplify. Remember to put in 0 for h after you factor out h from the numerator and cancel it to the h in the denominator. The remaining terms are your answer.

Jamison Newman

Beginner2022-04-24Added 10 answers

The limit definition tells us that:

y^{'} = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

where h is a small increment.
In our case we get:

lim_{h \to 0} \frac{[{(x + h)}^{2} + (x + h) + 1] - [x^{2} + x + 1]}{h} =

= lim_{h \to 0} \frac{x^{2} + 2 x h + h^{2} + x + h + 1 - x^{2} - x - 1}{h} =

lim_{h \to 0} (h \frac{2 x + h + 1}{h}) = lim_{h \to 0} (2 x + h + 1) =

h \to 0

we get:
y'=2x+1

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