অন্তরীকরণ-২ ( Differentiation-2 )

অনুশীলনী \(9.C / Q.2\)-এর প্রশ্নসমুহ

\(x\) কে পরিবর্তনশীল ধরে নিচের ফাংশনগুলির অন্তরজ নির্ণয় করঃ
\(Q.2.(i)\) \(\frac{x}{x^2+a^2}\)
উত্তরঃ \(\frac{a^2-x^2}{(x^2+a^2)^2}\)

\(Q.2.(ii)\) \(\frac{1+\sqrt{x}}{1-\sqrt{x}}\)
[ রাঃ ২০১১ ]
উত্তরঃ \(\frac{1}{\sqrt{x}(1-\sqrt{x})^2}\)

\(Q.2.(iii)\) \(\frac{x^2}{x^2-4}\)
উত্তরঃ \(-\frac{8x}{(x^2-4)^2}\)

\(Q.2.(iv)\) \(\frac{x^2+x+1}{x^2-x+1}\)
উত্তরঃ \(\frac{2(1-x^2)}{(x^2-x+1)^2}\)

\(Q.2.(v)\) \(\frac{1+x+x^2}{x}\)
উত্তরঃ \(\frac{x^2-1}{x^2}\)

\(Q.2.(vi)\) \(\frac{x}{e^x}\)
উত্তরঃ \( \frac{1-x}{e^x}\)

\(Q.2.(vii)\) \(\frac{1-\tan x}{1+\tan x}\)
[ বঃ ২০১৩; দিঃ২০১০ ]
উত্তরঃ \(\frac{-2\sec^2 x}{(1+\tan x)^2}\)

\(Q.2.(viii)\) \(\frac{x^n+\tan x}{e^x-\cot x}\)
\( \frac{(e^x-\cot x)(nx^{n-1}+\sec^2 x)-(x^n+\tan x)(e^x+cosec^2 \ x)}{(e^x-\cot x)^2}\)

\(Q.2.(ix)\) \(\frac{\tan x+\cot x}{3e^x}\)
\(\frac{(\tan x+\cot x)(\tan x-\cot x-1)}{3e^x}\)

\(Q.2.(x)\) \(\frac{\sin x}{x^2+\cos x}\)
\( \frac{1+x(x\cos x-2\sin x)}{(x^2+\cos x)^2}\)

\(Q.2.(xi)\) \(\frac{\log_ax}{x^2}\)
উত্তরঃ \(\frac{1-\ln x^2)}{x^3\ln a}\)

\(Q.2.(xii)\) \(\frac{x^{n}}{\log_ax}\)
উত্তরঃ \( \frac{x^{n-1}(n\ln x-1)}{\ln a(\log_ax)^2}\)

\(Q.2.(xiii)\) \(\frac{1-\ln x}{1+\ln x}\)
উত্তরঃ \(-\frac{2}{x(1+\ln x)^2}\)

\(Q.2.(xiv)\) \(\frac{\sec x}{1+\sec x}\)
উত্তরঃ \( \frac{\sin x}{(1+\cos x)^2}\)
\(Q.2.(xv)\) \(\frac{e^x-\tan x}{\ln x-\cot x}\)
\(\frac{(\ln x-\cot x)(e^x-\sec^2 x)-(e^x-\tan x)\left(\frac{1}{x}+cosec^2 \ x\right)}{(\ln x-\cot x)^2}\)

\(Q.2.(xvi)\) \(\frac{\ln x}{\cos x}\)
উত্তরঃ \( \frac{\cos x+x\ln x\sin x}{x\cos^2 x}\)

\(Q.2.(xvii)\) \(\frac{\sin x+\cos x}{\sin 2x}\)
উত্তরঃ \(\frac{1}{2}(\sec x\tan x-cosec \ x\cot x)\)

\(Q.2.(xviii)\) \(\frac{1+\sin x}{1+\cos x}\)
উত্তরঃ \( \frac{\cos x+\sin x+1}{(1+\cos x)^2}\)

\(Q.2.(xix)\) \(\frac{\sin x}{x}\)
উত্তরঃ \(\frac{x\cos x-\sin x}{x^2}\)

\(Q.2.(xx)\) \(\frac{x^4}{\ln x}\)
উত্তরঃ \( \frac{x^3(\ln x^4-1)}{(\ln x)^2}\)

\(Q.2.(xxi)\) \(\frac{\tan x-\cot x}{\tan x+\cot x}\)
উত্তরঃ \(2\sin 2x\)

\(Q.2.(xxii)\) \(\frac{a\sin x+b\cos x}{a\cos x-b\sin x}\)
উত্তরঃ \( \frac{a^2+b^2}{(a\cos x-b\sin x)^2}\)

\(Q.2.(xxiii)\) \(\frac{cosec \ x+\cot x}{cosec \ x-\cot x}\)
উত্তরঃ \(-\frac{2\sin x}{(1-\cos x)^2}\)

\(Q.2.(xxiv)\) \(\frac{e^x}{\ln x}\)
উত্তরঃ \( \frac{e^x(x\ln x-1)}{x(\ln x)^2}\)

\(Q.2.(xxv)\) \(\frac{\ln x}{1+\cos x}\)
উত্তরঃ \(\frac{1+\cos x+x\ln x\sin x}{x(1+\cos x)^2}\)

\(Q.2.(xxvi)\) \(\frac{e^x}{\cos x}\)
উত্তরঃ \( \frac{e^x(\sin x+\cos x)}{\cos^2 x}\)

\(Q.2.(xxvii)\) \(\frac{1-\cot x}{1+\cot x}\)
উত্তরঃ \(\frac{2cosec^2 \ x}{(1+\cot x)^2}\)

\(Q.2.(xxviii)\) \(\frac{x^n+\cot x}{e^x-\tan x}\)
\( \frac{(e^x-\tan x)(nx^{n-1}-cosec^2 \ x)-(x^n+\cot x)(e^x-\sec^2 x)}{(e^x-\tan x)^2}\)
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