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Differential Calculus> differentiate the following function with...

question mark

differentiate the following function with respect to x
(1)cos^-1(x/sqrt(x^2+a^2))
(2)sin^-1(sinx + cos x/sqrt(2)), -3pi/4
(3)cos^-1(cos x + sin x/sqrt(2)), -pi/4
(4)tan^-1(x/1+sqrt(1-x^2)), -1

taniska , 10 Years ago

Grade 12

2 Answers

Rinkoo Gupta

Last Activity: 10 Years ago


Y=cos-1x/v(x2+a2)


Dy/dx=-[1/v1-{x/v(x2+a2)}2].[{v(x2+a2).1-x.2x/v(x2+a2)}/(x2+a2)]


=-a/(x2+a2)
Thanks & Regards
Rinkoo Gupta
AskIITians Faculty

Arun Kumar

Last Activity: 10 Years ago

Hi

I'll solve only first question for the rest make another post.

$\\$assume u$=\frac{x}{x^2+a^2} \\\frac{d}{du}(cos^{-1}(u))=\frac{-1}{\sqrt{1-u^2}} \\\frac{d}{dx}(cos^{-1}(u))=\frac{-1}{\sqrt{1-u^2}}\frac{du}{dx} \\\frac{du}{dx}=\frac{\frac{d}{dx}\left(x\right)\left(x^2+a^2\right)-\frac{d}{dx}\left(x^2+a^2\right)x}{\left(x^2+a^2\right)^2} \\=\frac{a^2-x^2}{\left(a^2+x^2\right)^2}$

Substitue to get the answer.

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty

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