Example

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微分(2)

\( y = f(u) \quad u = g(v) \quad v = h(x) \) の場合の
\( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx} = f'(u) \cdot g'(v) \cdot h'(x)\) を計算すればよい．
\( y = \log | \cos( 2x+3 ) | \)
\( y = \log |u| \quad u = \cos(v) \quad v = 2x+3 \) に対応させて，
\( y = \log |u| \quad f'(u) = \frac{dy}{du} = \frac{1}{u} \)
\( u = \cos(v) \quad u'(v) = \frac{du}{dv} = -\sin(v) \)
\( v = 2x+3 \quad v'(x) = \frac{dv}{dx} = 2 \)
\( \frac{dy}{dx} = \frac{1}{u} \times -\sin(v) \times 2 = \frac{1}{\cos(2x+3)} \times -\sin(2x+3) \times 2\)
\( = -2\frac{\sin(2x+3)}{\cos(2x+3)}\)
\( = -2\tan(2x+3)\)

選択肢

関数
(1)	\(5 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\)	(2)	\(3 \cos{\left(3 x \right)}\)	(3)	\(6 \left(x - 2\right)^{5}\)
(4)	\(2 \left(x + 1\right) e^{x^{2} + 2 x + 1}\)	(5)	\(- 2 \sin{\left(2 x + 3 \right)}\)	(6)	\(\frac{3 \cos{\left(3 x \right)}}{x} - \frac{\sin{\left(3 x \right)}}{x^{2}}\)
(7)	\(- 6 x \left(2 - x\right)^{5} + \left(2 - x\right)^{6}\)	(8)	\(2 e^{2 x}\)	(9)	\(\frac{2}{\tan{\left(2 x + 3 \right)}}\)

学籍番号
氏　　名
微分を行い、解を選択肢から選びなさい．
(A) \(y = \left(2 - x\right)^{6}\)	\( \frac{dy}{dx} = \) ()
(B) \(y = \sin{\left(3 x \right)}\)	\( \frac{dy}{dx} = \) ()
(C) \(y = \cos{\left(2 x + 3 \right)}\)	\( \frac{dy}{dx} = \) ()
(D) \(y = \sin^{5}{\left(x \right)}\)	\( \frac{dy}{dx} = \) ()
(E) \(y = e^{2 x}\)	\( \frac{dy}{dx} = \) ()
(F) \(y = e^{\left(x + 1\right)^{2}}\)	\( \frac{dy}{dx} = \) ()
(G) \(y = \log{\left(- \sin{\left(2 x + 3 \right)} \right)}\)	\( \frac{dy}{dx} = \) ()