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Lösung Differentialquotient A

Zuletzt geändert von akukin am 2025/11/22 20:20

  1. \(\begin{align*} f'(1) &= \lim_{x \to 1} \frac{f(x)- f(1)}{x - 1} \\ &=\lim_{x \to 1} \frac{(x^2 + 3) - 4}{x - 1} \\ &= \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \\ &= \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1} \\ &= \lim_{x \to 1} (x + 1) \\ &= 1 + 1 \\ &= 2 \end{align*}\)

  2. \(\begin{align*} f'(1) &= \lim_{x \to 1} \frac{f(x)- f(1)}{x - 1} \\ &= \lim_{x \to 1} \frac{3x^2 - 3}{x - 1} \\ &= \lim_{x \to 1} \frac{3(x^2 - 1)}{x - 1} \\ &= \lim_{x \to 1} \frac{3(x - 1)(x + 1)}{x - 1} \\ &= \lim_{x \to 1} 3(x + 1) \\ &= 3(1 + 1) \\ &= 6 \end{align*}\)