Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 874

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\[1)\ f(x) = \sin^{3}x\]

\[u = \sin x\ f(u) = u^{3}:\]

\[f^{'}(x) = \left( \sin x \right)^{'} \bullet \left( u^{3} \right)^{'} =\]

\[= \cos x \bullet 3u^{2} = 3\sin^{2}x \bullet \cos x\]

\[2)\ f(x) = 8^{\cos x}\]

\[u = \cos x\ \ f(u) = 8^{u}:\]

\[f^{'}(x) = \left( \cos x \right)^{'} \bullet \left( 8^{u} \right)^{'} =\]

\[= - \sin x \bullet 8^{u} \bullet \ln 8 =\]

\[= - 8^{\cos x} \bullet \ln 8 \bullet \sin x\]

\[3)\ f(x) = \cos^{4}x\]

\[u = \cos x\ f(u) = u^{4}:\]

\[f^{'}(x) = \left( \cos x \right)^{'} \bullet \left( u^{4} \right)^{'} =\]

\[= - \sin x \bullet 4u^{3} =\]

\[= - 4 \bullet \cos^{3}x \bullet \sin x\]

\[4)\ f(x) = \ln\left( x^{3} \right)\]

\[u = x^{3}\ \ f(u) = \ln u:\]

\[f^{'}(x) = \left( x^{3} \right)^{'} \bullet \left( \ln u \right)^{'} =\]

\[= 3x^{2} \bullet \frac{1}{u} = \frac{3x^{2}}{x^{3}} = \frac{3}{x}.\]