Derivace funkce

9000063307

Část: 
C
Derivace funkce \(f\colon y =\ln \left(\cos 2x\right)\) je rovna:
\(f'(x) = -2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 2\mathop{\mathrm{tg}}\nolimits 2x,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = -2,\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (-\frac{\pi }{4} + k\pi ; \frac{\pi } {4} + k\pi \right )\)
\(f'(x) = 1 -\ln\left(\sin 2x\right),\ x\in \mathop{\mathop{\bigcup }}\nolimits _{k\in \mathbb{Z}}\left (k\pi ; \frac{\pi } {2} + k\pi \right )\)

9000063109

Část: 
B
Derivace funkce \(f\colon y = 3^{x}\cdot x^{3}\) je rovna:
\(f'(x) = 3^{x}x^{2}(x\ln 3 + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x+1}x^{2}\ln 3,\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x\ln x + 3),\ x\in \mathbb{R}^{+}\)

9000063108

Část: 
B
Derivace funkce \(f\colon y = x^{5}\mathrm{e}^{x}\) je rovna:
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x),\ x\in \mathbb{R}\)
\(f'(x) = 5x^{4}\mathrm{e}^{x},\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(x - 5),\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x^{2}),\ x\in \mathbb{R}\)

9000063105

Část: 
B
Derivace funkce \(f\colon y = \frac{\sqrt{x}-1} {\sqrt{x}+1}\) je rovna:
\(f'(x) = \frac{1} {\sqrt{x}(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{\sqrt{x}} {(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{2} {x(\sqrt{x}+1)^{2}} ,\ x > 0\)
\(f'(x) = \frac{1} {(\sqrt{x}+1)^{2}} ,\ x > 0\)

9000063110

Část: 
B
Derivace funkce \(f\colon y =\sin x(1 +\mathop{\mathrm{tg}}\nolimits x)\) je rovna:
\(f'(x) =\cos x +\sin x + \frac{\sin x} {\cos ^{2}x},\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =\cos x +\sin x,\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) = \frac{\sin x} {\cos ^{2}x},\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)
\(f'(x) =\cos x + 2\sin x,\ x\in \mathbb{R}\setminus\{\frac{\pi}{2}+k\pi; k\in \mathbb{Z}\}\)