Example

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微分(2)

\( u, v \)　が \( x \) の関数で微分可能であり，\( k, a \) が定数とすると
\( \left( k u \right)' = k u' \)
\( \left( u \pm v\right)' = u' \pm v' \)
\( \left( u v\right)' = u' v + u v' \)
\( \Large \left( \frac{u}{v} \right)' \normalsize = \left( u v^{-1}\right)' = u' v^{-1} + u \left( v^{-1} \right)'\)
\( = u' v^{-1} + u \times -v^{-2} \times v'\)
\( = \Large \frac{u' v - u v'}{v^2} \)

学籍番号
氏　　名
微分を行い、解を選択肢から選びなさい．
(A) \(y = \log{\left(\cos{\left(x \right)} \right)}\)	\( \frac{dy}{dx} = \) ()
(B) \(y = x \sin{\left(x \right)}\)	\( \frac{dy}{dx} = \) ()
(C) \(y = \frac{1}{\sin{\left(x \right)}}\)	\( \frac{dy}{dx} = \) ()
(D) \(y = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1}\)	\( \frac{dy}{dx} = \) ()
(E) \(y = \left(x + 1\right) \left(x^{2} + 5 x + 2\right)\)	\( \frac{dy}{dx} = \) ()
(F) \(y = \left(x - 1\right) \left(x^{2} + 1\right) \left(x^{3} - 2 x\right)\)	\( \frac{dy}{dx} = \) ()
(G) \(y = a x + b\)	\( \frac{dy}{dx} = \) ()
(H) \(y = a x^{b}\)	\( \frac{dy}{dx} = \) ()
(I) \(y = \tan{\left(x \right)}\)	\( \frac{dy}{dx} = \) ()
(J) \(y = \frac{\tan^{2}{\left(x \right)}}{2}\)	\( \frac{dy}{dx} = \) ()
(K) \(y = a e^{x}\)	\( \frac{dy}{dx} = \) ()
(L) \(y = 2 \tanh{\left(x \right)}\)	\( \frac{dy}{dx} = \) ()
(M) \(y = \frac{\tanh^{2}{\left(x \right)}}{2}\)	\( \frac{dy}{dx} = \) ()
(N) \(y = \log{\left(a x \right)}\)	\( \frac{dy}{dx} = \) ()
(O) \(y = \frac{\log{\left(x \right)}}{\log{\left(a \right)}}\)	\( \frac{dy}{dx} = \) ()
(P) \(y = \log{\left(\cosh{\left(x \right)} \right)}\)	\( \frac{dy}{dx} = \) ()
(Q) \(y = \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}}\)	\( \frac{dy}{dx} = \) ()

関数に対数が含まれる場合，\(x\)は対数の定義範囲とする．
\( \log(x) \)は底が\( e \)の対数とする．
\( \log_a(x) = \frac{\log(x)}{\log(a)}\)
双曲関数
\( \sinh( x ) = \frac{e^{x} - e^{-x}}{2} \)
\( \cosh( x ) = \frac{e^{x} + e^{-x}}{2} \)
\( \tanh( x ) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \)

選択肢

関数
(1)	\(x \cos{\left(x \right)} + \sin{\left(x \right)}\)	(2)	\(\frac{1}{\sin^{2}{\left(x \right)}}\)	(3)	\(\cos{\left(x \right)}\)
(4)	\(6 x^{5} - 5 x^{4} - 4 x^{3} + 3 x^{2} - 4 x + 2\)	(5)	\(a b x^{b - 1}\)	(6)	\(\frac{2}{\cosh^{2}{\left(x \right)}}\)
(7)	\(\frac{\tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}\)	(8)	\(- \tan{\left(x \right)}\)	(9)	\(\frac{1}{\cos^{2}{\left(x \right)}}\)
(10)	\(\tanh{\left(x \right)}\)	(11)	\(\frac{1}{x \log{\left(a \right)}}\)	(12)	\(\frac{1}{a x}\)
(13)	\(- \frac{1}{\sin{\left(x \right)} + 1}\)	(14)	\(- \cos{\left(x \right)}\)	(15)	\(\sin{\left(x \right)}\)
(16)	\(x e^{x - 1}\)	(17)	\(a e^{x}\)	(18)	\(\frac{\tanh{\left(x \right)}}{\cosh^{2}{\left(x \right)}}\)
(19)	\(\frac{1}{\cosh^{2}{\left(x \right)}}\)	(20)	\(- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}\)	(21)	\(a\)
(22)	\(3 x^{2} + 12 x + 7\)	(23)	\(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}\)	(24)	\(\frac{1}{x}\)