חדו״א — כיתה י"ב · 5 יח"ל (קשה)

פתרונות

1. $2x\ln x+x$ — כלל המכפלה: $f'(x)=2x\ln x+x^2\cdot\frac{1}{x}=2x\ln x+x$.
2. $x^2 e^x(3+x)$ — $f'(x)=3x^2 e^x+x^3 e^x=x^2 e^x(3+x)$.
3. $e^x(3x+1)$ — $f'(x)=3e^x+(3x-2)e^x=e^x(3+3x-2)=e^x(3x+1)$.
4. $e^x\left(\frac{1}{2\sqrt{x}}+\sqrt{x}\right)$ — $f'(x)=\frac{1}{2\sqrt{x}}e^x+\sqrt{x}\,e^x=e^x\left(\frac{1}{2\sqrt{x}}+\sqrt{x}\right)$.
5. $\frac{x^2-2x}{(x-1)^2}$ — $\frac{2x(x-1)-x^2\cdot 1}{(x-1)^2}=\frac{2x^2-2x-x^2}{(x-1)^2}=\frac{x^2-2x}{(x-1)^2}$.
6. $\frac{e^x(x-1)}{x^2}$ — $\frac{e^x\cdot x-e^x\cdot 1}{x^2}=\frac{e^x(x-1)}{x^2}$.
7. $\frac{1-\ln x}{x^2}$ — $\frac{\frac{1}{x}\cdot x-\ln x\cdot 1}{x^2}=\frac{1-\ln x}{x^2}$.
8. $\frac{-2}{(x-1)^2}$ — $\frac{1\cdot(x-1)-(x+1)\cdot 1}{(x-1)^2}=\frac{x-1-x-1}{(x-1)^2}=\frac{-2}{(x-1)^2}$.
9. $\frac{1-x}{e^x}$ — $\frac{1\cdot e^x-x\cdot e^x}{(e^x)^2}=\frac{e^x(1-x)}{e^{2x}}=\frac{1-x}{e^x}$.
10. $4$ — $f'(x)=\frac{4(x^2+1)-4x\cdot 2x}{(x^2+1)^2}=\frac{4-4x^2}{(x^2+1)^2}$. בנקודה $x=0$: $\frac{4}{1}=4$.