Szczegółowe wyjaśnienie:
[tex]A)\ f(x)=\ln\left(\dfrac{6x}{x+6}\right)\\\\f'(x)=\dfrac{1}{\frac{6x}{x+6}}\cdot\dfrac{6(x+6)-6x\cdot1}{(x+6)^2}=\dfrac{x+6}{6x}\cdot\dfrac{6x+36-6x}{(x+6)^2}\\\\=\dfrac{(x+6)\cdot36}{6x(x+6)^2}=\dfrac{6(x+6)}{x(x+6)^2}=\dfrac{6}{x(x+6)}=\dfrac{6}{x^2+6x}[/tex]
skorzystałem z:
[tex]\left(\ln x\right)'=\dfrac{1}{x}\\\\\bigg[f\bigg(g(x)\bigg)\bigg]'=f'\bigg(g(x)\bigg)\cdot g'(x)\\\\\bigg[\dfrac{f(x)}{g(x)}\bigg]'=\dfrac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}[/tex]
[tex]B) f(x)=x^6\cos6x\\\\f'(x)=6x^5\cos6x+x^6(-\sin6x)\cdot6=6x^5\cos6x-6x^6\sin6x[/tex]
skorzystałem z:
[tex]\bigg[f(x)\cdot g(x)\bigg]'=f'(x)g(x)+f(x)g'(x)\\\\(\cos x)'=-\sin x\\\\\bigg[f\bigg(g(x)\bigg)\bigg]'=f'\bigg(g(x)\bigg)\cdotg'(x)[/tex]