65. From first principle prove that $$\frac{d}{d x}\left(x^n\right)=n x^{n-1}$$, where $$n$$ is a fixed number or rational.
66. From the first principal prove that $$\frac{d}{d(x)}(\sin x)=\cos x$$
67. Find the derivative of each of the following the first principle.
(i) $$(a x+b)$$
(ii) $$x^8$$
(iii) $$\frac{1}{x^3}$$
68. If $$u, v, w$$ are differentiable functions of $$x$$, prove that
$$
\frac{d}{d x}(u v w)=(u v) \cdot \frac{d w}{d x}+(w u) \cdot \frac{d u}{d x}+(w v) \cdot \frac{d u}{d x}
$$
69. Differentiate $$x^n \cot x$$
70. Differentiate $$\left(x^2+2 x-3\right)\left(x^2+7 x+5\right)$$
71. Differentiate $$(\tan x+\sec x)(\cot x+{cosec} x)$$
72. Differentiate $$\left(\frac{x^2+3 x-1}{x+2}\right)$$
73. Differentiate $$\left(\frac{5 x^2+6 x+7}{2 x^2+3 x+4}\right)$$
74. Differentiate $$\left(\frac{e^x+\sin x}{1+\log x}\right)$$
75. Differentiate $$\left(\frac{a x^2+b x+c}{p x^2+q x+r}\right)$$
76. Differentiate with respect to $$x \cos (\sin \sqrt{a x+b})$$.
77. Differentiate with respect to $$x: e^{2 x} \sin 3 x$$.
78. Differentiate with respect to $$x: \frac{e^{2 x}+x^3}{{cosec} 2 x}$$.
79. Find the derivative of $$f(x)=\frac{x}{1+\tan x}$$.
80. Find the derivative of $$\frac{1}{a x^2+b}$$, with respect to $$x$$.
81. Find derivative of $$f(x)=\frac{1}{x}$$ by detail method.