"How do you use the limit definition of..." solved and explained

Trent Fuller

Answered question

2022-04-11

How do you use the limit definition of the derivative to find the derivative of

f (x) = 2 x^{2} - 3 x + 6

tralhavahr9c

Beginner2022-04-12Added 16 answers

Given a function f(x), the derivative of f(x) is defined as:

\frac{d f (x)}{d x} = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

f (x) = 2 x^{2} - 3 x + 6

then

f (x + h) = 2 {(x + h)}^{2} - 3 (x + h) + 6

= 2 x^{2} + 2 x h + h^{2} - 3 x - 3 h + 6

f (x + h) - f (x) = 2 x^{2} + 2 x h + h^{2} - 3 x - 3 h + 6

= \frac{- (2 x^{2} - 3 x + 6)}{2 x h + h^{2} + 3 h}

and

\frac{f (x + h) - f (x)}{h} = 2 x + h + 3

Therefore

\frac{d f (x)}{d x} = lim_{h \to 0} 2 x + h + 3 = 2 x + 3

phoenixtreeaung

Beginner2022-04-13Added 19 answers

Evaluate the limit of the expression

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

Explanation:
So

lim_{h \to 0} \frac{(2 {(x + h)}^{2} - 3 (x + h) + 6) - f (3 x^{2} - 3 x + 6)}{h}

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