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Question Number 195137 by mathlove last updated on 25/Jul/23

$$f\left(x\right)=arctan\left(\frac{4sinx}{3+5cosx}\right)then{f}^{{}^{\prime }}\left(\frac{\pi }{3}\right)=?$$

Answered by Tokugami last updated on 02/Sep/23

$$f^{\prime }\left(x\right)=\frac{1}{1+{\left(\frac{4\sin x}{3+5\cos x}\right)}^2}\frac{d}{dx}\left(\frac{4\sin x}{3+5\cos x}\right)$$$$=\frac{1}{1+\frac{{16\sin }^{2}x}{{(3+5\cos x)}^2}}\frac{(3+5\cos x)\left(4\cos x\right)-\left(4\sin x\right)(-5\sin x)}{{(3+5\cos x)}^2}$$$$=\frac{12\cos x+{20\cos }^{2}x+{20\sin }^{2}x}{{(3+5\cos x)}^2+{16\sin }^{2}x}$$$$=\frac{12\cos x+20}{9+30\cos x+{25\cos }^{2}x+{16\sin }^{2}x}$$$$=\frac{4(3\cos x+5)}{25+30\cos x+{9\cos }^{2}x}=\frac{4\cancel{(3\cos x+5)}}{{(3\cos x+5)}^{\cancel{2}}}$$$$=\frac{4}{5+3\cos x}$$$$f^{\prime }\left(\frac{\pi }{3}\right)=\frac{4}{5+3\cos \left(\frac{\pi }{3}\right)}=\frac{4}{5+3\left(\frac{1}{2}\right)}=\frac{4}{5+\frac{3}{2}}$$$$f^{\prime }\left(\frac{\pi }{3}\right)=\frac{8}{13}$$$$$$

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