Question 28.6.1: Differentiate: (i) e^x/x (ii)(2x+3/x²-5) (iii) e^x/(1+sin x)...
Differentiate:
(i) \frac{e^{x}}{x} \qquad(ii) \left(\frac{2 x+3}{x^{2}-5}\right)\qquad (iii) \frac{e^{x}}{(1+\sin x)}
(i) \frac{d}{d x}\left(\frac{e^{x}}{x}\right)=\frac{x \cdot \frac{d}{d x}\left(e^{x}\right)-e^{x} \cdot \frac{d}{d x}(x)}{x^{2}}=\frac{x e^{x}-e^{x} \cdot 1}{x^{2}}=\frac{e^{x}(x-1)}{x^{2}} .
(ii) \frac{d}{d x}\left(\frac{2 x+3}{x^{2}-5}\right)=\frac{\left(x^{2}-5\right) \cdot \frac{d}{d x}(2 x+3)-(2 x+3) \cdot \frac{d}{d x}\left(x^{2}-5\right)}{\left(x^{2}-5\right)^{2}} \\ \\=\frac{\left(x^{2}-5\right) \cdot 2-(2 x+3) \cdot 2 x}{\left(x^{2}-5\right)^{2}}=\frac{-2\left(x^{2}+3 x+5\right)}{\left(x^{2}-5\right)^{2}} \text {. }\\
(iii) \frac{d}{d x}\left(\frac{e^{x}}{1+\sin x}\right)=\frac{(1+\sin x) \cdot \frac{d}{d x}\left(e^{x}\right)-e^{x} \cdot \frac{d}{d x}(1+\sin x)}{(1+\sin x)^{2}} \\ \\=\frac{(1+\sin x) \cdot e^{x}-e^{x}(\cos x)}{(1+\sin x)^{2}}=\frac{(1+\sin x-\cos x) e^{x}}{(1+\sin x)^{2}} .