Differentiate each the following from first principles : (i) 2x (ii) 1

1.	Differentiate each the following from first principles : (i) 2x (ii) 1√x (iii) 1x3 (iv) x2+1x (v) x2−1x (vi) x+1x+2 (vii) x+23x+5 (viii) k xn (ix) 1√3−x (x) x2+x+3 (xi) (x+2)3 (xii) x3+4x2+3x+2 (xiii) (x2+1)(x−5) (xiv) √2x2+1 (xv) 2x+3x−2
Answer» Differentiate each the following from first principles : $(i) \frac{2}{x}$ $(i i) \frac{1}{\sqrt{x}}$ $(i i i) \frac{1}{x^{3}}$ $(i v) \frac{x^{2} + 1}{x}$ $(v) \frac{x^{2} - 1}{x}$ $(v i) \frac{x + 1}{x + 2}$ $(v i i) \frac{x + 2}{3 x + 5}$ $(v i i i) k x^{n}$ $(i x) \frac{1}{\sqrt{3 - x}}$ $(x) x^{2} + x + 3$ $(x i) (x + 2)^{3}$ $(x i i) x^{3} + 4 x^{2} + 3 x + 2$ $(x i i i) (x^{2} + 1) (x - 5)$ $(x i v) \sqrt{2 x^{2} + 1}$ $(x v) \frac{2 x + 3}{x - 2}$

Differentiate each the following from first principles : (i) 2x (ii) 1√x (iii) 1x3 (iv) x2+1x (v) x2−1x (vi) x+1x+2 (vii) x+23x+5 (viii) k xn (ix) 1√3−x (x) x2+x+3 (xi) (x+2)3 (xii) x3+4x2+3x+2 (xiii) (x2+1)(x−5) (xiv) √2x2+1 (xv) 2x+3x−2

Answer»

Differentiate each the following from first principles :

$(i) \frac{2}{x}$

$(i i) \frac{1}{\sqrt{x}}$

$(i i i) \frac{1}{x^{3}}$

$(i v) \frac{x^{2} + 1}{x}$

$(v) \frac{x^{2} - 1}{x}$

$(v i) \frac{x + 1}{x + 2}$

$(v i i) \frac{x + 2}{3 x + 5}$

$(v i i i) k x^{n}$

$(i x) \frac{1}{\sqrt{3 - x}}$

$(x) x^{2} + x + 3$

$(x i) (x + 2)^{3}$

$(x i i) x^{3} + 4 x^{2} + 3 x + 2$

$(x i i i) (x^{2} + 1) (x - 5)$

$(x i v) \sqrt{2 x^{2} + 1}$

$(x v) \frac{2 x + 3}{x - 2}$