AvgStudent
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- Jan 1, 2022
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Find [imath]\boxed{\frac{d}{dx}(\sin \sqrt{x})}[/imath] using the definition of derivative.
Hint: Use the following facts
[math]\boxed{1)\,\sin(A)-\sin(B)=2\sin\Bigg(\frac{A-B}{2}\Bigg)⋅\cos\Bigg(\frac{A+B}{2}\Bigg)}\\ \boxed{2)\, \lim_{\theta \to 0}\,\frac{\sin\theta}{\theta}=1}[/math]What I have so far...
[math]\lim_{h \to 0} \frac{\sin(\sqrt{x}+h)- \sin{\sqrt{x}}}{h}= 2\cdot\lim_{h \to 0} \frac{\sin(\frac{\sqrt{x+h}-\sqrt{x}}{2})\cos(\frac{\sqrt{x+h}+\sqrt{x}}{2})}{h}[/math]I'm supposed to use the second fact here somehow but don't see it. Help, please
Hint: Use the following facts
[math]\boxed{1)\,\sin(A)-\sin(B)=2\sin\Bigg(\frac{A-B}{2}\Bigg)⋅\cos\Bigg(\frac{A+B}{2}\Bigg)}\\ \boxed{2)\, \lim_{\theta \to 0}\,\frac{\sin\theta}{\theta}=1}[/math]What I have so far...
[math]\lim_{h \to 0} \frac{\sin(\sqrt{x}+h)- \sin{\sqrt{x}}}{h}= 2\cdot\lim_{h \to 0} \frac{\sin(\frac{\sqrt{x+h}-\sqrt{x}}{2})\cos(\frac{\sqrt{x+h}+\sqrt{x}}{2})}{h}[/math]I'm supposed to use the second fact here somehow but don't see it. Help, please
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