53. Differentiate the following functions w. r. t. x.
(i) $$\frac{1-e^x}{1+e^x}$$
(ii) $$\frac{\log x}{e^x}$$
(iii) $$\frac{\log x-4 x^3}{\log x+4 x^3}$$
(iv) $$\frac{\mathrm{e}^{\mathrm{x}}}{1+\cos \mathrm{x}}$$
54. A particle moves along a line so that at time $$t$$, its position is $$\mathrm{s}(\mathrm{t})=6 \mathrm{t}-\mathrm{t}^2$$.
55. A particle moves along a line so that its position at time $$t$$ is $$\mathrm{s}(\mathrm{t})=\frac{\mathrm{t}^2+2}{\mathrm{t}+1} \quad$$ units. Find its velocity at times $$\mathrm{t}=3$$.
56. Differentiate coefficient of $$\cos x$$ w.r.t., $$x$$ from first principles Let $$\mathrm{y}=\cos \mathrm{x}$$.
57. Find the differential coefficients of the following from first principles.
(i) $$5 x^4$$
(ii) $$\frac{1}{\sqrt{2 x}}$$
(iii) $$\mathrm{x}^3-27$$
58. Find the derivative, $$3 \sin \mathrm{x}+2 \sin \alpha$$, where $$\alpha$$ is a constant.
59. Differentiate the following functions w.r.t. to $$x$$., $$\mathrm{x}^3+7 \mathrm{x}+3+4 \mathrm{a}^{2 \mathrm{a}}+5 \mathrm{a}^2$$, where a is a constant.
60. Find the slope of the tangent to the curve $$\mathrm{f}(\mathrm{x})=2 \mathrm{x}^6+\mathrm{x}^4-1$$ at $$\mathrm{x}=1$$.
61. If $$f(x)=x^2-9 x+20$$, then find $$f^{\prime}(x)$$ and hence find $$\mathrm{f}^{\prime}(100)$$ and $$\mathrm{f}^{\prime}\left(\frac{9}{2}\right)$$.
62. If $$y=\frac{2-3 \cos x}{\sin x}$$, find $$\frac{d y}{d x}$$ at $$x=\frac{\pi}{4}$$
63. If $$y=(-x-a)(x-b)$$, then find the value of $$x$$ for which $$\frac{\mathrm{dy}}{\mathrm{dx}}=0$$.
64. Differentiate the following functions w.r.t. to x .
(i) $$\left(a x^2+\sin x\right)(p+q \cos x)$$
(ii) $$x^4(5 \sin x-3 \cos x)$$