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Question Number 152310 by DELETED last updated on 27/Aug/21

Answered by DELETED last updated on 27/Aug/21

$$1).\mathrm{f}\left(\mathrm{x}\right)={({\mathrm{x}}^2-\mathrm{x})}^{10}$$$$\mathrm{simbol}={\mathrm{f}}^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{\mathrm{dy}}{\mathrm{dx}}=....?$$$$\mathrm{misal}:\mathrm{U}={\mathrm{x}}^2-\mathrm{x}\to \frac{\mathrm{dU}}{\mathrm{dx}}=2\mathrm{x}-1$$$$\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{nU}}^{\mathrm{n}-1}.\frac{\mathrm{dU}}{\mathrm{dx}}$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=10{({\mathrm{x}}^2-\mathrm{x})}^{10-1}.(2\mathrm{x}-1)$$$$=10(2\mathrm{x}-1){({\mathrm{x}}^2-\mathrm{x})}^9$$$$=(20\mathrm{x}-10){({\mathrm{x}}^2-\mathrm{x})}^9//$$

Answered by DELETED last updated on 27/Aug/21

$$2).\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{{\mathrm{x}}^4+5}$$$$\mathrm{Bentuk}\mathrm{soal}=\frac{\mathrm{U}}{\mathrm{V}}$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{U}}^{\prime }\mathrm{V}-{\mathrm{UV}}^{\prime }}{{\mathrm{V}}^2}$$$$\mathrm{U}=1\to {\mathrm{U}}^{\prime }=0$$$$\mathrm{V}={\mathrm{x}}^4+5\to {\mathrm{V}}^{\prime }={4\mathrm{x}}^3$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{U}}^{\prime }\mathrm{V}-{\mathrm{UV}}^{\prime }}{{\mathrm{V}}^2}$$$$=\frac{0.({\mathrm{x}}^4+5)-1.{4\mathrm{x}}^3}{{({\mathrm{x}}^4+5)}^2}$$$$=\frac{{4\mathrm{x}}^3}{{({\mathrm{x}}^4+5)}^2}//$$

Answered by DELETED last updated on 27/Aug/21

$$3).\mathrm{Bentuk}\mathrm{soal}:\frac{\mathrm{U}}{\mathrm{V}}$$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{6\mathrm{x}}{\sqrt{2\mathrm{x}-1}}$$$$\mathrm{U}=6\mathrm{x}\to {\mathrm{U}}^{\prime }=6$$$$\mathrm{V}=\sqrt{2\mathrm{x}-1}={(2\mathrm{x}-1)}^{\frac{1}{2}}$$$${\mathrm{V}}^{\prime }=\frac{1}{2}{(2\mathrm{x}-1)}^{\frac{1}{2}-1}.\left(2\right)$$$$={(2\mathrm{x}-1)}^{-\frac{1}{2}}=\frac{1}{{(2\mathrm{x}-1)}^{\frac{1}{2}}}$$$$=\frac{1}{\sqrt{2\mathrm{x}-1}}$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{U}}^{\prime }\mathrm{V}-{\mathrm{UV}}^{\prime }}{{\mathrm{V}}^2}$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{6.\sqrt{2\mathrm{x}-1}-6\mathrm{x}.\frac{1}{\sqrt{2\mathrm{x}-1}}}{{\left(\sqrt{2\mathrm{x}-1}\right)}^2}$$$$=\frac{6.\sqrt{2\mathrm{x}-1}-\frac{6\mathrm{x}}{\sqrt{2\mathrm{x}-1}}}{2\mathrm{x}-1}$$$$$$$$=\frac{\frac{6(2\mathrm{x}-1)-6\mathrm{x}}{\sqrt{2\mathrm{x}-1}}}{2\mathrm{x}-1}$$$$=\frac{\frac{12\mathrm{x}-6-6\mathrm{x}}{\sqrt{2\mathrm{x}-1}}}{2\mathrm{x}-1}$$$$=\frac{\frac{6\mathrm{x}-6}{\sqrt{2\mathrm{x}-1}}}{2\mathrm{x}-1}$$$$=\frac{6(\mathrm{x}-1)}{(2\mathrm{x}-1)\sqrt{2\mathrm{x}-1}}//$$$$$$

Answered by DELETED last updated on 27/Aug/21

$$4).\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\frac{\mathrm{x}+2}{\mathrm{x}-1}}={\left[\frac{\mathrm{x}+2}{\mathrm{x}-1}\right]}^{\frac{1}{2}}$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{n}{\left[\frac{\mathrm{U}}{\mathrm{V}}\right]}^{\mathrm{n}-1}.\left(\frac{{\mathrm{U}}^{\prime }\mathrm{V}-{\mathrm{UV}}^{\prime }}{{\mathrm{V}}^2}\right)$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{n}{\left[\frac{\mathrm{U}}{\mathrm{V}}\right]}^{\mathrm{n}-1}.\left(\frac{{\mathrm{U}}^{\prime }\mathrm{V}-{\mathrm{UV}}^{\prime }}{{\mathrm{V}}^2}\right)$$$$\mathrm{U}=2\mathrm{x}+2\to {\mathrm{U}}^{\prime }=2$$$$\mathrm{V}=\mathrm{x}-1\to {\mathrm{V}}^{\prime }=1$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}{\left[\frac{2\mathrm{x}+2}{\mathrm{x}-1}\right]}^{\frac{1}{2}-1}.\left(\frac{2.(\mathrm{x}-1)-(2\mathrm{x}+2).1}{{(\mathrm{x}-1)}^2}\right)$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}{\left[\frac{2\mathrm{x}+2}{\mathrm{x}-1}\right]}^{-\frac{1}{2}}.\left(\frac{2\mathrm{x}-2-2\mathrm{x}-2)}{{(\mathrm{x}-1)}^2}\right)$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}{\left[\frac{\mathrm{x}-1}{2\mathrm{x}+2}\right]}^{\frac{1}{2}}.\left(\frac{-4}{{(\mathrm{x}-1)}^2}\right)$$$$\frac{\mathrm{dy}}{\mathrm{dx}}={\left[\frac{\mathrm{x}-1}{2\mathrm{x}+2}\right]}^{\frac{1}{2}}.\left(\frac{-2}{{(\mathrm{x}-1)}^2}\right)$$$$\frac{\mathrm{dy}}{\mathrm{dx}}={\left[\frac{\mathrm{x}-1}{2\mathrm{x}+2}\right]}^{\frac{1}{2}}\left(\frac{-2}{{(\mathrm{x}-1)}^2}\right)=-\frac{2}{{(2\mathrm{x}+2)}^{\frac{1}{2}}{(\mathrm{x}-1)}^{\frac{3}{2}}}$$$$=-\frac{2}{(\mathrm{x}-1)\sqrt{(2\mathrm{x}+2)(\mathrm{x}-1)}}//$$$$$$

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