Get the answer to "Let f(x)=−3x3+9x+4, how do you use the limit..."

jermandcryspza3

Answered question

2022-03-22

Let

f (x) = - 3 x^{3} + 9 x + 4

, how do you use the limit definition of the derivative to calculate the derivative of f?

Melody Gamble

Beginner2022-03-23Added 10 answers

The definition is:

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

.
So:

lim_{h \to 0} \frac{- 3 {(x + h)}^{3} + 9 (x + h) + 4 - (- 3 x^{3} + 9 x + 4)}{h}

= lim_{h \to 0} \frac{- 3 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + 9 x + 9 h + 4 + 3 x^{3} - 9 x - 4}{h} =

= lim_{h \to 0} \frac{- 3 x^{3} - 9 x^{2} h - 9 x h^{2} - 3 h^{3} + 9 x + 9 h + 4 + 3 x^{3} - 9 x - 4}{h} =

= lim_{h \to 0} \frac{- 9 x^{2} h - 9 x h^{2} - 3 h^{3} + 9 h}{h} =

= lim_{h \to 0} (- 9 x^{2} - 9 x h - 3 h^{2} + 9) = - 9 x^{2} + 9,

that is the derivative of the function.

ron4d3ozip7

Beginner2022-03-24Added 9 answers

The derivative of f(x) with respect to x is defined as:

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

= lim_{h \to 0} (\frac{- 3 {(x + h)}^{3} + 9 (x + h) + 4 - (- 3 x^{3} + 9 x + 4)}{h})

= lim_{h \to 0} (\frac{- 3 {(x + h)}^{3} + 9 (x + h) + 4 - (- 3 x^{3} + 9 x + 4)}{h}) = - 3 (3 x^{2} - 3)

Answer:

= - 3 (3 x^{2} - 3)

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