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Grade 12Differential Calculus

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

taniska

12 Years agoGrade 12

2 Answers

Arun Kumar

12 Years ago

Hi

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

applying chain rule.

$\\\frac{d}{dx}\left(\arcsin\left(\frac{1}{\sqrt{1+x^2}}\right)\right) \\=\frac{d}{du}\left(\arcsin \left(u\right)\right)\frac{d}{dx}\left(\frac{1}{\sqrt{1+x^2}}\right) \\=\frac{1}{\sqrt{1-u^2}}\left(-\frac{x}{\left(x^2+1\right)^{\frac{3}{2}}}\right) \\=\frac{1}{\sqrt{1-\left(\frac{1}{\sqrt{1+x^2}}\right)^2}}\left(-\frac{x}{\left(x^2+1\right)^{\frac{3}{2}}}\right) \\=-\frac{\text{sgn}\left(x\right)}{x^2+1} \\&sgn(x)=\frac{x}{|x|}&$

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

Rinkoo Gupta

12 Years ago

Y=sin^-1{1/v(1+x²)}

Dy/dx=[1/v1-{1/v(1+x²)}²].{-(1+x²)^-3/2/2}

On simplifying we get

Dy/dx=-1/(1+x²)

Thanks & Regards

Rinkoo Gupta

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