Use the definition of derivative (four step method) to calculate 𝑓 ′ (−1) for 𝑓(𝑥) = 𝑥^3 − 2𝑥.

Answer 1

The definition of a derivative is lim h→0 (f(x+h)−f(x))/h.

Step 1:

f(x+h) = (x+h)³ - 2(x+h) and f(x) = x³-2x

Step 2:

f(x+h) - f(x) = (x+h)³ - 2(x+h) - x³ + 2x = x³+3x²h+3xh²+h³ - 2x-2h -x³ + 2x

= 3x²h + 3xh² + h³ - 2h

Step 3: Divide out h

(f(x+h) - f(x))/h = 3x² + 3xh + h² - 2

Step 4: Take the limit, lim h→0, set h = 0 here:

f'(x) = 3x² -2

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f'(-1) = 3(-1)² - 2 = 3 - 2 = 1