Differentiate from the first principles:
(i) $2 x$ (ii) ${(x - 1)}^{2}$

Open in App

Solution

$(i) - 2 x$
$f^{'} (x) =_{h \to 0} \frac{2 (x + h) - f (x)}{h}$
$= lim h \to 0 \frac{2 (x + h) - 2 x}{h}$
$= lim h \to 0 \frac{2 x + 2 h - 2 x}{h}$
$= lim h \to 0 \frac{2 h}{h}$
$lim h \to 0 2$
$= 2$

$(i i)$ $(x - 1)^{2}$
$f^{'} (x) =_{x \to 0} + \frac{x + h) - f (x)}{h}$
$= lim h \to 0 + \frac{(x + h - 1)^{2} - (x - 1)^{2}}{h}$
$= lim h \to 0 + \frac{x + h - 1 + x - 1) (x + h - 1 - x + 1)}{h}$
$= lim h \to 0 + \frac{(2 x - 2 + h) (h)}{h}$
$= 2 (x - 1)$

Suggest Corrections

Similar questions

(- x)^{- 1}

c o s (x - \frac{π}{8})

Differentiate each of the following from first principles :

$(i) t a n^{2} x$

$(i i) t a n (2 x + 1)$

$(i i i) t a n 2 x$

$(i i v) \sqrt{t a n x}$

e^{\sqrt{\cot x}}

(i) s i n \sqrt{2} x

(i i) c o s \sqrt{x}

(i i i) t a n \sqrt{x}

(i v) t a n x^{2}

Differentiate each the following from first principles :

$(i) x s i n x$

$(i i) x + c o s x$

$(i i i) s i n (2 x - 3)$

$(i v) \sqrt{s i n 2 x}$

$(v) \frac{s i n x}{x}$

$(v i) \frac{c o s x}{x}$

$(v i i) x^{2} s i n x$

$(v i i i) \sqrt{s i n (3 x + 1)}$

$(i x) s i n x + c o s x$

Join BYJU'S Learning Program