Differentiate following w.r.t x Exercise 1.1 XI

  Differentiate the following w.r.t x Exercise 1.1

3)Differentiate the following w.r.t x :

(xi)$\frac{e^{\sqrt{x}}+1}{e^{\sqrt{x}}-1}$

Solution:

$\frac{dy}{dx}=\frac{d}{dx}\left(\frac{e^{\sqrt{x}}+1}{e^{\sqrt{x}}-1}\right)$

$=\frac{(e^{\sqrt{x}}-1)\frac{d}{dx}(e^{\sqrt{x}}+1)-(e^{\sqrt{x}}+1)\frac{d}{dx}(e^{\sqrt{x}}-1)}{(e^{\sqrt{x}}-1)^2}$

$=\frac{(e^{\sqrt{x}}-1)(e^{\sqrt{x}}\frac{d}{dx}\sqrt{x}+0)-(e^{\sqrt{x}}+1)(e^{\sqrt{x}}\frac{d}{dx}\sqrt{x}-0)}{(e^{\sqrt{x}}-1)^2}$

$=\frac{(e^{\sqrt{x}}-1)(e^{\sqrt{x}}\frac{1}{2\sqrt{x}})-(e^{\sqrt{x}}+1)(e^{\sqrt{x}}\frac{1}{2\sqrt{x}})}{(e^{\sqrt{x}}-1)^2}$

(ii)$(1+4x)^5(3x+x-x^2)^8$

(iii)$\frac{x}{\sqrt{7-3x}}$

(iv)$\frac{(x^3-5)^5}{(x^3+5)^3}$

(v)$(1+sin^2x)^2+(1+cos^2)^3$

(vi)$\sqrt{cos x}+\sqrt{cos\sqrt{x}}$

(vii)$log(sec 3x+tan 3x)$

(viii)$\frac{1+sin x^{\circ}}{1-sin x^{\circ}}$

(x)$\frac{e^{2x}-e^{-2x}}{e^{2x}+e{-2x}}$

(xi)$\frac{e^{\sqrt{x}}+1}{e^{\sqrt{x}}-1}$

(xii)$log\left[ tan^3xsin^4x(x^2+7)\right]$

(xiii)$log\left[\sqrt{\frac{1-cos3x}{1+cos3x}}\right]$

(xiv)$log\left[\sqrt{\frac{1+cos\frac{5x}{2}}{1-cos\frac{5x}{2}}}\right]$

(xv)$log\left[\sqrt{\frac{1-sinx}{1+sinx}}\right]$

Differentiate following w.r.t x Exercise 1.1 XIl