1.4 KiB
Wyznaczyć pochodne podanych funkcji.
1
y=(3-x^{4})^{2}
y'=2(3-x^{4})\cdot(-4x^{3})=(6-2x^{4})\cdot(-4x^{3})=-24x^{3}+8x^{7}
2
y=1-\frac{4}{x^6}+\frac{3}{x^7}
y'=-4\cdot-6x^{-7}+3\cdot-7x^{-8}=\frac{24}{x^{7}}-\frac{21}{x^{8}}
3
y=\cfrac{x^{2}+4}{x}
y'=\cfrac{1}{x^{2}}\cdot\left(2x^{2}-(x^{2}+4)\right)=\cfrac{x^{2}-4}{x^{2}}=1-\cfrac{4}{x^2}
4
y=\cfrac{x-3}{x^{2}}
y'=\cfrac{1}{x^{4}}\cdot\left(x^{2}-2x^{2}+6x\right)=\cfrac{6x-x^{2}}{x^{4}}=\cfrac{6-x}{x^3}
5
y=(\sqrt[4]{x}-\frac{1}{2})^{2}
y=(x^{\frac{1}{4}}-\frac{1}{2})^{2}
y'=2(\sqrt[4]{x}-\frac{1}{2})\cdot\frac{1}{4}{x}^{-\frac{3}{4}}=2(\sqrt[4]{x}-\frac{1}{2})\cdot\frac{1}{16\sqrt{x^{3}}}=\frac{2\sqrt[4]{x}-1}{16\sqrt{x^{3}}}
6
7
y=\cfrac{x^{5}+x^{3}+x}{\sqrt{x}}
y'=\cfrac{1}{x}\left((x^{5}+x^{3}+x)\cdot\frac{1}{2\sqrt{x}}-(5x^{4}+3x^2+1)\cdot\sqrt{x}\right)
8
y=\sqrt[3]{\sqrt{x}x}
y'=\frac-{1}{3}
9
10
y=x\cos x
y'=\cos x \cdot x-\sin x
11
y=x^{2}\sin x + \tan x
y'=2x(\sin x)\cdot x^2\cos x+\frac{1}{\cos^{2}x}
12
y=\cfrac{\sin x}{x^3}
y'=\frac{1}{x^{6}}\cdot \left(\sin x\cdot 3x^{2}-\cos x\cdot x^{3}\right)
y'=\cfrac{\left(\sin x\cdot 3x^{2}-\cos x\cdot x^{3}\right)}{x^6}
y'=\frac{3\sin x}{x^4}-\frac{\cos x}{x^{3}}
13
y=\sqrt{x}\ln x
y'=-\cfrac{1}{2\sqrt{x}}\cdot \ln x + \cfrac{1}{x}\cdot\sqrt{x}
y'=\cfrac{\sqrt{x}}{x}-\cfrac{\ln x}{2\sqrt{x}}