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Question Number 203158 by hardmath last updated on 11/Jan/24

$$1.\mathrm{y}={\mathrm{tg}}^2\mathrm{x}\Rightarrow {\mathrm{y}}^{{}^{\prime }}=?$$$$2.\underset{\mathbf{x}\to 0}{\lim }\frac{\ln (1+2\mathrm{x})}{\sin 2\mathrm{x}}=?$$

Answered by shunmisaki007 last updated on 11/Jan/24

$$1.\mathrm{Are}t,g,\mathrm{and}x\mathrm{all}\mathrm{variable}\mathrm{or}\mathrm{one}\mathrm{or}\mathrm{two}\mathrm{of}\mathrm{them}\mathrm{constant}?$$$$2.\mathrm{Because}\underset{x\to 0}{\lim }\ln (1+2x)=0$$$$\mathrm{and}\underset{x\to 0}{\lim }\sin \left(2x\right)=0,$$$$\mathrm{by}{\mathrm{L}}^{\prime }{\mathrm{hopital}}^{\prime }\mathrm{s}\mathrm{rule},$$$$\underset{x\to 0}{\lim }\frac{\ln (1+2x)}{\sin \left(2x\right)}=\underset{x\to 0}{\lim }\frac{\frac{d}{dx}\left(\ln (1+2x)\right)}{\frac{d}{dx}\left(\sin \left(2x\right)\right)}$$$$=\underset{x\to 0}{\lim }\frac{\frac{2}{1+2x}}{2\cos \left(2x\right)}$$$$=1.★$$

Commented by Frix last updated on 11/Jan/24

$$tgx=\tan x$$$${tg}^{2}x={\tan }^{2}x={\left(\tan x\right)}^2$$

Answered by witcher3 last updated on 11/Jan/24

$$=\underset{x\to 0}{\lim }\frac{\ln (1+2\mathrm{x})}{2\mathrm{x}}.\frac{1}{\frac{\sin \left(2\mathrm{x}\right)}{2\mathrm{x}}}=1.\frac{1}{1}=1$$

Answered by shunmisaki007 last updated on 11/Jan/24

$$1.y={\tan }^2\left(x\right)$$$$y^{\prime }=2\tan \left(x\right){\sec }^2\left(x\right)=2\tan \left(x\right)+{2\tan }^3\left(x\right)★$$

Commented by Frix last updated on 11/Jan/24

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