Differentiate following w.r.t x Exercise 1.1 x

  Differentiate the following w.r.t x Exercise 1.1

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

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

Solution- $\frac{dy}{dx}=\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$

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

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

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

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

(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 X