Derivácia funkcie

Úloha 34. Vypočítajte deriváciu funkcie $$f(x)=-2\arctan{\sqrt{\frac{3-x}{x-1}}}.$$

Riešenie:
$$f^{\prime}(x)= -2\frac{1}{1+\left(\frac{3-x}{x-1}\right)}\cdot \frac{1}{2}\left(\frac{3-x}{x-1}\right)^{-\frac{1}{2}}\frac{1-x+x-3}{(x-1)^2}=$$
$$-\frac{2(x-1)}{x-1+3-x}\cdot \frac{1}{2}\sqrt{\frac{x-1}{3-x}}\cdot\frac{-2}{(x-1)^2}= -\frac{2(x-1)}{2}\cdot \frac{1}{2}\sqrt{\frac{x-1}{3-x}}\cdot\frac{-2}{(x-1)^2}=$$
$$\sqrt{\frac{x-1}{3-x}}\cdot\frac{1}{x-1}= \sqrt{\frac{1}{(3-x)(x-1)}}.$$

Úloha 35. Vypočítajte deriváciu funkcie $$g(x)=\ln\left(e^x+\sqrt{e^{2x}-1}\right)+\arctan(e^{2x}-1).$$

Riešenie:
$$g^{\prime}(x)=\frac{1}{e^x+\sqrt{e^{2x}-1}}\cdot \left(e^x+\frac{2e^{2x}}{2\sqrt{e^{2x}-1}}\right)+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}=$$
$$\frac{1}{e^x+\sqrt{e^{2x}-1}}\cdot \left(e^x+\frac{e^{2x}}{\sqrt{e^{2x}-1}}\right)+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}=$$

$$\frac{1}{e^x+\sqrt{e^{2x}-1}}\cdot \left(\frac{e^x\sqrt{e^{2x}-1}}{\sqrt{e^{2x}-1}}+\frac{e^{2x}}{\sqrt{e^{2x}-1}}\right)+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}=$$
$$\frac{1}{e^x+\sqrt{e^{2x}-1}}\cdot \frac{e^x\sqrt{e^{2x}-1}+e^{2x}}{\sqrt{e^{2x}-1}}+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}=$$
$$\frac{1}{e^x+\sqrt{e^{2x}-1}}\cdot \frac{e^x(\sqrt{e^{2x}-1}+e^x)}{\sqrt{e^{2x}-1}}+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}= \frac{e^x}{\sqrt{e^{2x}-1}}+\frac{2e^{2x}}{1+\left(e^{2x}-1\right)^2}.$$