Expert Answer to "Using the limit definition, how do you find..."

yert21trey123z7

Answered question

2022-04-20

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

f (x) = - 5 x^{2} + 8 x + 2

Answer & Explanation

utloverej

Beginner2022-04-21Added 15 answers

Using the limit definition to find the derivative of f(x)

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

so it can be expressed as following

lim_{h \to 0} \frac{(- 5 {(x + h)}^{2} + 8 (x + h) + 2) - (- 5 x^{2} + 8 x + 2)}{h}

=-10x+8

mandamosppy

Beginner2022-04-22Added 21 answers

$f^{'} (x) = lim_{h \to 0} \frac{- 5 {(x + h)}^{2} + 8 (x + h) + 2 - (- 5 x^{2} + 8 x + 2)}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{- 5 (x^{2} + 2 x h + h^{2}) + 8 x + 8 h + 2 + 5 x^{2} - 8 x - 2}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{- 5 x^{2} - 10 x h - 5 h^{2} + 8 x + 8 h + 2 + 5 x^{2} - 8 x - 2}{h}$
$f^{'} (x) = lim_{h \to 0} \frac{- 10 x h - 5 h^{2} + 8 h}{h}$
$f' (x) = \lim_{h \to 0} \frac{h (- 10 x h - 5 h + 8)}{h}$
f'(x)=-10x-5(0)+8
f'(x)=-10x+8
$∴$ The derivative is $\frac{d y}{d x} = - 10 x + 8$ .

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