find derivative of :- e root cot x by first principle - Maths - Limits and Derivatives

Question

find derivative of :-
e root cot x
by first principle ?

Lalit Mehra · Accepted Answer

Dear Student!

$Let f (x) = e^{\sqrt{\cot x}} ∴ f (x + h) = e^{\sqrt{\cot (x + h)}}$

We know that,

$f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} (First Principle o f d i f f e r e n t i a t i o n) ∴ f' (x) = \lim_{h \to 0} \frac{e^{\sqrt{\cot (x + h)}} {- e}^{\sqrt{\cot x}}}{h} \Rightarrow f' (x) = \lim_{h \to 0} \frac{e^{\sqrt{\cot x} [\frac{e^{\sqrt{\cot (x + h)}}}{e^{\sqrt{\cot x}}} - 1]}}{h} \Rightarrow f' (x) = \lim_{h \to 0} \frac{e^{\sqrt{\cot x} [e^{\sqrt{\cot (x + h)} - \sqrt{\cot x}} - 1]}}{h} \Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} (\frac{e^{\sqrt{\cot (x + h)} - \sqrt{\cot x}} - 1}{\sqrt{\cot (x + h)} - \sqrt{\cot} x}) \times \frac{[\sqrt{\cot (x - h)} - \sqrt{\cot} x)]}{h}$

$\Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} (\frac{e^{\sqrt{\cot (x + h)} - \sqrt{\cot x}} - 1}{\sqrt{\cot (x + h)} - \sqrt{\cot} x}) \times \lim_{h \to 0} \frac{(\sqrt{\cot (x + h)} - \sqrt{\cot} x)}{h} \Rightarrow f' (x) = e^{\sqrt{\cot x}} \times 1 \times \lim_{h \to 0} \frac{\cot (x + h) - \cot x}{h (\sqrt{\cot (x + h)} + \sqrt{\cot x})} (∵ \lim_{x \to 0} \frac{e^{x} - 1}{x} = 1)$

$\Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} \frac{(\frac{\cos (x + h)}{\sin (x + h)} - \frac{\cos x}{\sin x})}{h (\sqrt{\cot (x + h)} + \sqrt{\cot x})} \Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} \frac{(\sin x \cos (x + h) - \cos x \sin (x + h))}{h \sin (x + h) \sin x (\sqrt{\cot (x + h)} + \sqrt{\cot x})} \Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} \frac{\sin [x - (x + h)]}{h \sin (x + h) \sin x \sqrt{\cot (x + h)} + \sqrt{\cot x}}$

$\Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} \frac{\sin (- h)}{h \sin (x + h) \sin x (\sqrt{\cot (x + h)} + \sqrt{\cot x})} \Rightarrow f' (x) = e^{\sqrt{\cot x}} \lim_{h \to 0} \frac{- \sin h}{h \sin (x + h) \sin x (\sqrt{\cot (x + h)} + \sqrt{\cot x})} \Rightarrow f' (x) = - \frac{e^{\sqrt{\cot x}} \times (\lim_{h \to 0} \frac{\sin h}{h})}{\lim_{h \to 0} \sin (x + h) \sin x (\sqrt{\cot (x + h)} + \sqrt{\cot x})} \Rightarrow f' (x) = - \frac{e^{\sqrt{\cot x}} \times 1}{\sin x \times \sin x (\sqrt{\cot x} + \sqrt{\cot} x)} \Rightarrow f' (x) = \frac{{- e}^{\sqrt{\cot x}}}{\sin^{2} x 2 \sqrt{\cot} x} \Rightarrow f' (x) = \frac{{- e}^{\sqrt{\cot x}} {cosec}^{2} x}{2 \sqrt{\cot} x}$

Cheers!

find derivative of :- e root cot x by first principle ?

find derivative of :-

e ^{root cot x}

by first principle ?