જો y = operatorname{Tanh}^{-1} sqrt{frac{1-x} | vedclass

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(B) આપેલ છે $y = \operatorname{Tanh}^{-1} \sqrt{\frac{1-x}{1+x}}$.ધારો કે $x = \cos 	heta$,તેથી $	heta = \cos^{-1} x$.તેથી $\sqrt{\frac{1-x}{1+x}} =

Vedclass · Accepted Answer

$\frac{-1}{2 x \sqrt{1-x^2}}$

Vedclass · Answer

(B) આપેલ છે $y = \operatorname{Tanh}^{-1} \sqrt{\frac{1-x}{1+x}}$.ધારો કે $x = \cos 	heta$,તેથી $	heta = \cos^{-1} x$.તેથી $\sqrt{\frac{1-x}{1+x}} = \sqrt{\frac{1-\cos 	heta}{1+\cos 	heta}} = \sqrt{\frac{2 \sin^2 (	heta/2)}{2 \cos^2 (	heta/2)}} = 	an(	heta/2)$.તેથી,$y = \operatorname{Tanh}^{-1} (	an(	heta/2))$.નિત્યસમ $\operatorname{Tanh}^{-1} z = \frac{1}{2} \ln \left( \frac{1+z}{1-z} ight)$ નો ઉપયોગ કરતા:$y = \frac{1}{2} \ln \left( \frac{1+	an(	heta/2)}{1-	an(	heta/2)} ight) = \frac{1}{2} \ln \left( 	an \left( \frac{\pi}{4} + \frac{	heta}{2} ight) ight) = \frac{1}{2} \left( \ln 	an \left( \frac{\pi}{4} + \frac{\cos^{-1} x}{2} ight) ight)$.$x$ ની સાપેક્ષ વિકલન કરતા:$\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{	an(\frac{\pi}{4} + \frac{	heta}{2})} \cdot \sec^2(\frac{\pi}{4} + \frac{	heta}{2}) \cdot \frac{1}{2} \cdot \frac{d}{dx}(\cos^{-1} x)$.$\frac{dy}{dx} = \frac{1}{4} \cdot \frac{1}{\sin(\frac{\pi}{4} + \frac{	heta}{2}) \cos(\frac{\pi}{4} + \frac{	heta}{2})} \cdot \left( -\frac{1}{\sqrt{1-x^2}} ight)$.$\frac{dy}{dx} = \frac{1}{2 \sin(\frac{\pi}{2} + 	heta)} \cdot \left( -\frac{1}{\sqrt{1-x^2}} ight) = \frac{1}{2 \cos 	heta} \cdot \left( -\frac{1}{\sqrt{1-x^2}} ight) = -\frac{1}{2x \sqrt{1-x^2}}$.

જો $y = \operatorname{Tanh}^{-1} \sqrt{\frac{1-x}{1+x}}$ હોય,તો $\frac{dy}{dx} = $

Similar Questions

જો $y = \sin^{-1}(\frac{2x}{1 + x^2})$ હોય,તો $\left. \frac{dy}{dx} \right|_{x = -2}$ ની કિંમત શોધો.

$\frac{d}{dx} \left\{ \sin^2 \left( \cot^{-1} \sqrt{\frac{1 + x}{1 - x}} \right) \right\} =$

જો $2y = {\left( {{{\cot }^{ - 1}}\left( {\frac{{\sqrt 3 \cos x + \sin x}}{{\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}$ અને $x \in \left( {0,\frac{\pi }{2}} \right)$ હોય,તો $\frac{{dy}}{{dx}}$ ની કિંમત શોધો.

$\frac{d}{dx} \left[ \tan^{-1} \left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \right] = $

જો $y = \tan^{-1} \left( \frac{\sqrt{1+x^2}-1}{x} \right)$ હોય, તો $y'(1)$ ની કિંમત શોધો.

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