To find the derivative of f(x) = −2x³ using the first principle, we use the definition: f'(x) = lim_(h → 0) (f(x+h) − f(x))/(h) Step 1: Find f(x+h). f(x+h) = −2(x+h)³ Expand (x+h)³: (x+h)³ = x³ + 3x²h + 3xh² + h³ Substitute this back into f(x+h): f(x+h) = −2(x³ + 3x²h + 3xh² + h³) = −2x³ − 6x²h − 6xh² − 2h³ Step 2: Calculate f(x+h) − f(x). f(x+h) − f(x) = (−2x³ − 6x²h − 6xh² − 2h³) − (−2x³); f(x+h) − f(x) = −2x³ − 6x²h − 6xh² − 2h³ + 2x³; f(x+h) − f(x) = −6x²h − 6xh² − 2h³ Step 3: Divide by h. (f(x+h) − f(x))/(h) = (−6x²h − 6xh² − 2h³)/(h) Factor out h from the numerator: (h(−6x² − 6xh − 2h²))/(h) = −6x² − 6xh − 2h² Step 4: Take the limit as h → 0. f'(x) = lim_(h → 0) (−6x² − 6xh − 2h²); Substitute h=0 into the expression: f'(x) = −6x² − 6x(0) − 2(0)²; f'(x) = −6x² − 0 − 0; f'(x) = −6x² The final answer is −6x²

First principle

October 1, 2026Mathematics

ScanSolve AI Answer

Answer

-6x^2

Step-by-step solution

4 steps

To find the derivative of $f(x) = -2x^3$ using the first principle, we use the definition:

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

Find $f(x+h)$ .
$f(x+h) = -2(x+h)^3$
Expand $(x+h)^3$ :
$(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$
Substitute this back into $f(x+h)$ :
$f(x+h) = -2(x^3 + 3x^2h + 3xh^2 + h^3) = -2x^3 - 6x^2h - 6xh^2 - 2h^3$
Calculate $f(x+h) - f(x)$ .
$f(x+h) - f(x) = (-2x^3 - 6x^2h - 6xh^2 - 2h^3) - (-2x^3)$ $f(x+h) - f(x) = -2x^3 - 6x^2h - 6xh^2 - 2h^3 + 2x^3$ $f(x+h) - f(x) = -6x^2h - 6xh^2 - 2h^3$
Divide by $h$ .
$\frac{f(x+h) - f(x)}{h} = \frac{-6x^2h - 6xh^2 - 2h^3}{h}$
Factor out $h$ from the numerator:
$\frac{h(-6x^2 - 6xh - 2h^2)}{h} = -6x^2 - 6xh - 2h^2$
Take the limit as $h \to 0$ .
$f'(x) = \lim_{h \to 0} (-6x^2 - 6xh - 2h^2)$
Substitute $h=0$ into the expression:
$f'(x) = -6x^2 - 6x(0) - 2(0)^2$ $f'(x) = -6x^2 - 0 - 0$ $f'(x) = -6x^2$
The final answer is $-6x^2$

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First principle

Answer

Step-by-step solution

Find f(x+h)f(x+h)f(x+h).

Calculate f(x+h)−f(x)f(x+h) - f(x)f(x+h)−f(x).

Divide by hhh.

Take the limit as h→0h \to 0h→0.

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More Mathematics Questions

Find $f(x+h)$ .

Calculate $f(x+h) - f(x)$ .

Divide by $h$ .

Take the limit as $h \to 0$ .