3. Oblicz pochodną i uprość wynik

[Januszgolenia]

\(y= \frac{1+sin(2x)}{cos(2x)}\)

[britva]

\(\left(\dfrac{\sin\left(2\,x\right)+1}{\cos\left(2\,x\right)}\right)'_{x}=\dfrac{2\,\sin\left(2\,x\right)\,\left(\sin\left(2\,x\right)+1\right)+2\,\cos^{2}\left(2\,x\right)}{\cos^{2}\left(2\,x\right)}=\dfrac{2\,\sin^{2}\left(2\,x\right)+2\,\sin\left(2\,x\right)}{\cos^{2}\left(2\,x\right)}+2\)

krok po kroku

[eresh]

\(y= \frac{1+sin(2x)}{cos(2x)}\)

\(f'(x)=\frac{2\cos 2x\cdot \cos 2x+2\sin 2x(1+\sin 2x)}{\cos^22x}\\
f'(x)=\frac{2\cos^22x+2\sin 2x+2\sin^22x}{\cos^22x}\\
f'(x)=\frac{2(1+\sin 2x)}{\cos^22x}\\
f'(x)=\frac{2(1+\sin 2x)}{1-\sin^22x}\\
f'(x)=\frac{2(1+\sin 2x)}{(1-\sin 2x)(1+\sin 2x)}\\
f'(x)=\frac{2}{1+\sin 2x}\)