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Lösung Differentialquotient B

Zuletzt geändert von akukin am 2025/11/22 20:23

  1. \(\begin{align*} f'(x_0) &= \lim_{x \to x_0} \frac{f(x)- f(x_0)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{(x^2 + 3) - (x_0^2 + 3)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{x^2 - x_0^2}{x - x_0} \\ &= \lim_{x \to x_0} \frac{(x - x_0)(x + x_0)}{x - x_0} \\ &= \lim_{x \to x_0} (x + x_0) \\ &= x_0 + x_0 \\ &= 2x_0 \end{align*}\)

  2. \(\begin{align*} f'(x_0) &= \lim_{x \to x_0} \frac{f(x)- f(x_0)}{x - x_0}\\ &= \lim_{x \to x_0} \frac{3x^2 - 3x_0^2}{x - x_0} \\ &= \lim_{x \to x_0} \frac{3(x^2 - x_0^2)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{3(x - x_0)(x + x_0)}{x - x_0} \\ &= \lim_{x \to x_0} 3(x + x_0) \\ &= 3(2x_0) \\ &= 6x_0 \end{align*}\)