Szczegółowe wyjaśnienie:
[tex]1.\\f(x)=\left(12x^5+9x^4-\dfrac{9}{x}+2\ln x\right)^5=\left(12x^5+9x^4-9x^{-1}+2\ln x\right)^5\\\\f'(x)=5\left(12x^5+9x^4-9x^{-1}+2\ln x\right)^4\cdot\left(5\cdot12x^4+4\cdot9x^3-(-1)\cdot9x^{-2}+2\cdot\dfrac{1}{x}\right)\\\\=5\left(12x^5+9x^4-\dfrac{9}{x}+2\ln x\right)^4\cdot\left(60x^4+36x^3+\dfrac{9}{x^2}+\dfrac{2}{x}\right)[/tex]
Skorzystałem ze wzorów:
[tex]\bigg[f\bigg(g(x)\bigg)\bigg]'=f'\bigg(g(x)\bigg)\cdot g'(x)\\\\\bigg(x^n\bigg)'=nx^{x-1}\\\\\left(\ln x\right)'=\dfrac{1}{x}[/tex]
[tex]2.\\f(x)=\text{arctg}\sqrt{7x^3+8x}\\\\f'(x)=\left(\text{arctg}\sqrt{7x^3+8x}\right)'\cdot\left(\sqrt{7x^3+8x}\right)'\cdot\left(7x^3+8x\right)'\\\\=\dfrac{1}{1+\left(\sqrt{7x^3+8x}\right)^2}\cdot\dfrac{1}{2\sqrt{7x^3+8x}}\cdot(3\cdot7x^2+8)\\\\=\dfrac{21x^2+8}{2\sqrt{7x^3+8x}+2\left(\sqrt{7x^3+8x}\right)^3}[/tex]
Skorzystałem ze wzorów:
[tex]\bigg[f\bigg(g(x)\bigg)\bigg]'=f'\bigg(g(x)\bigg)\cdot g'(x)\\\\\left(\text{arctg}x\right)'=\dfrac{1}{1+x^2}\\\\\left(\sqrt{x}\right)'=\dfrac{1}{2\sqrt{x}}\\\\\left(x^n\right)^n=nx^{n-1}[/tex]
