লগারিদমের সাহায্যে অন্তরীকরণ (Differentiation By Logarithm)

অনুশীলনী \(9.E / Q.1\)-এর প্রশ্নসমুহ
নিচের ফাংশনগুলির \(x\)-এর সাপেক্ষে অন্তরজ নির্ণয় কর।
\(Q.1.(i)\) \(\ln{[x-\sqrt{x^2-1}]}\)
উত্তরঃ \(-\frac{1}{\sqrt{x^2-1}}\)
[ কুঃ২০১০,২০০৩;চঃ২০০৫;রাঃ২০০০;মাঃ২০০৬]

\(Q.1.(ii)\) \(\ln{[\sqrt{x-a}+\sqrt{x-b}]}\)
উত্তরঃ \(\frac{1}{2\sqrt{(x-a)(x-b)}}\)
[ চঃ২০০১]

\(Q.1.(iii)\) \(\ln{(x+\sqrt{x^2+a^2})}\)
উত্তরঃ \(\frac{1}{\sqrt{x^2+a^2}}\)
[ চঃ২০১৭]

\(Q.1.(iv)\) \(\ln{(ax^2+bx+c)}\)
উত্তরঃ \(\frac{2ax+b}{ax^2+bx+c}\)

\(Q.1.(v)\) \(\ln{\left(\frac{1+\sqrt{x}}{1-\sqrt{x}}\right)}\)
উত্তরঃ \(\frac{1}{\sqrt{x}(1-x)}\)

\(Q.1.(vi)\) \(\ln{\left(\frac{x^2+x+1}{x^2-x+1}\right)}\)
উত্তরঃ \(\frac{2(1-x^2)}{1+x^2+x^4}\)

\(Q.1.(vii)\) \(\ln{\left(\sqrt{\frac{1-\cos{x}}{1+\cos{x}}}\right)}\)
উত্তরঃ \( cosec{x}\)
[ ঢাঃ২০১২,২০০৭;রাঃ২০১১]

\(Q.1.(viii)\) \(\ln{\{e^x\left(\frac{x-1}{x+1}\right)^{\frac{3}{2}}\}}\)
উত্তরঃ \(\frac{x^2+2}{x^2-1}\)
[ সিঃ২০০৩]

\(Q.1.(ix)\) \(\ln{\left(\frac{1+x}{1-x}\right)^{\frac{1}{4}}}-\frac{1}{2}\tan^{-1}{x}\)
উত্তরঃ \(\frac{x^2}{1-x^4}\)
[ দিঃ২০১০]

\(Q.1.(x)\) \(\log_{x}a\)
উত্তরঃ \(-\frac{\ln{a}}{x(\ln{x})^2}\)
[ যঃ২০১৭,২০০৮;চঃ২০০৬;রাঃ২০০১]

\(Q.1.(xi)\) \(\log_{x}2x\)
উত্তরঃ \(-\frac{\ln{2}}{x(\ln{x})^2}\)
\(Q.1.(xii)\) \(\log_{\cos{x}}\tan{x}\)
উত্তরঃ \(\frac{cosec{x}\sec{x}\ln{(\cos{x})}+\tan{x}\ln{\tan{x}}}{(\ln{\cos{x}})^2}\)

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

\(Q.1.(xiv)\) \(\frac{e^{-x}(3x+5)}{7x-1}\)
উত্তরঃ \(-e^{-x}\left(\frac{21x^2+32x+33}{(7x-1)^2}\right)\)
[ যঃ২০০৫]

\(Q.1.(xv)\) \(\frac{(x+1)^2\sqrt{x-1}}{(x+4)^3e^x}\)
উত্তরঃ \(\frac{(x+1)^2\sqrt{x-1}}{(x+4)^3e^x}\left(\frac{2}{x+1}+\frac{1}{2(x-1)}+\frac{3}{x+4}-1\right)\)
[ কুঃ২০০৯]

\(Q.1.(xvi)\) \(x^3\sqrt{\frac{x^2+4}{x^2+3}}\)
উত্তরঃ \(x^3\sqrt{\frac{x^2+4}{x^2+3}}\left(\frac{3}{x}+\frac{x}{x^2+4}-\frac{x}{x^2+3}\right)\)

\(Q.1.(xvii)\) \(\frac{x\cos^{-1}{x}}{\sqrt{1-x^2}}\)
উত্তরঃ \(\frac{x\cos^{-1}{x}}{\sqrt{1-x^2}}\left(\frac{1}{x}-\frac{1}{\sqrt{1-x^2}\cos^{-1}{x}}+\frac{x}{1-x^2}\right)\)

\(Q.1.(xviii)\) \(\frac{e^{x^2}\tan^{-1}{x}}{\sqrt{1+x^2}}\)
উত্তরঃ \(\frac{e^{x^2}\tan^{-1}{x}}{\sqrt{1+x^2}}\left(2x+\frac{1}{(1+x^2)\tan^{-1}{x}}-\frac{x}{1+x^2}\right)\)

\(Q.1.(xix)\) \(\frac{(x^2+1)^2}{\sqrt[3]{x^2}}\)
উত্তরঃ \(\frac{2}{3}\left(5x^{\frac{7}{3}}+4x^{\frac{1}{3}}-x^{-\frac{5}{3}}\right)\)

\(Q.1.(xx)\) \(\left(\frac{x}{1+\sqrt{1-x^2}}\right)^{n}\)
উত্তরঃ \(\frac{n}{x\sqrt{1-x^2}}\left(\frac{x}{1+\sqrt{1-x^2}}\right)^{n}\)

\(Q.1.(xxi)\) \(\frac{x\log{x}}{\sqrt{1+x^2}}\)
উত্তরঃ \(\frac{x\log{x}}{\sqrt{1+x^2}}\left(\frac{1}{x}+\frac{\log_{10}e}{x\log{x}}-\frac{x}{1+x^2}\right)\)
[ কুয়েটঃ ২০০৫-২০০৬]
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