Determine as expressões designatórias das funções derivadas

1. a)
   $$\begin{array}{*{20}{l}}
   {f'(x)}& = &{\left( {\operatorname{sen} (3x) + \cos x} \right)’} \\
   {}& = &{\cos (3x) \times \left( {3x} \right)’ – \operatorname{sen} x} \\
   {}& = &{3\cos (3x) – \operatorname{sen} x}
   \end{array}$$
   b)
   $$\begin{array}{*{20}{l}}
   {g'(x)}& = &{\left( {{{\cos }^2}(2x)} \right)’} \\
   {}& = &{2 \times \cos (2x) \times \left( {\cos (2x)} \right)’} \\
   {}& = &{2\cos (2x) \times \left( { – \operatorname{sen} (2x) \times \left( {2x} \right)’} \right)} \\
   {}& = &{ – 2\left( {2\cos (2x) \times \operatorname{sen} (2x)} \right)} \\
   {}& = &{ – 2\operatorname{sen} (4x)}
   \end{array}$$
   c)
   \[\begin{array}{*{20}{l}}
   {h’\left( \alpha  \right)}& = &{{{\left( {\frac{{1 – \cos (3\alpha )}}{\alpha }} \right)}^\prime }} \\
   {}& = &{\frac{{\operatorname{sen} (3\alpha ) \times {{\left( {3\alpha } \right)}^\prime } \times \alpha  – 1 \times \left( {1 – \cos (3\alpha )} \right)}}{{{\alpha ^2}}}} \\
   {}& = &{\begin{array}{*{20}{l}}
   {\frac{{3\alpha \operatorname{sen} (3\alpha ) – 1 + \cos (3\alpha )}}{{{\alpha ^2}}},}&{\alpha  \ne 0}
   \end{array}}
   \end{array}\]
   d)
   \[\begin{array}{*{20}{l}}
   {i’\left( z \right)}& = &{{{\left( {\frac{{1 – \cos (2z)}}{{1 + \cos (2z)}}} \right)}^\prime }} \\
   {}& = &{\frac{{\operatorname{sen} (2z) \times {{\left( {2z} \right)}^\prime } \times \left( {1 + \cos (2z)} \right) + \operatorname{sen} (2z) \times {{\left( {2z} \right)}^\prime } \times \left( {1 – \cos (2z)} \right)}}{{{{\left( {1 + \cos (2z)} \right)}^2}}}} \\
   {}& = &{\frac{{2\operatorname{sen} (2z) + 2\operatorname{sen} (2z)\cos (2z) + 2\operatorname{sen} (2z) – 2\operatorname{sen} (2z)\cos (2z)}}{{{{\left( {1 + \cos (2z)} \right)}^2}}}} \\
   {}& = &{\begin{array}{*{20}{l}}
   {\frac{{4\operatorname{sen} (2z)}}{{{{\left( {1 + \cos (2z)} \right)}^2}}},}&{\alpha  \ne \frac{\pi }{2} + k\pi ,k \in \mathbb{Z}}
   \end{array}}
   \end{array}\]
   e)
   $$\begin{array}{*{20}{l}}
   {j'(t)}& = &{\left( {\cos \left( {4 – 3t} \right)} \right)’} \\
   {}& = &{ – \operatorname{sen} \left( {4 – 3t} \right) \times \left( {4 – 3t} \right)’} \\
   {}& = &{3\operatorname{sen} \left( {4 – 3t} \right)}
   \end{array}$$
   ­
2. a)
   \[\begin{array}{*{20}{l}}
   {{{\left( {f\left( {g(x)} \right)} \right)}^\prime }\left( 1 \right)}& = &{f’\left( {g\left( 1 \right)} \right) \times g’\left( 1 \right)} \\
   {}& = &{f’\left( 3 \right) \times 2} \\
   {}& = &{5 \times 2} \\
   {}& = &{10}
   \end{array}\]
   b)
   \[\begin{array}{*{20}{l}}
   {{{\left( {f\left( {g\left( {2x + 1} \right)} \right)} \right)}^\prime }\left( 0 \right)}& = &{f’\left( {g\left( {2 \times 0 + 1} \right)} \right) \times g’\left( {2 \times 0 + 1} \right) \times {{\left( {2x + 1} \right)}^\prime }} \\
   {}& = &{f’\left( {g\left( 1 \right)} \right) \times g’\left( 1 \right) \times 2} \\
   {}& = &{f’\left( 3 \right) \times 2 \times 2} \\
   {}& = &{5 \times 2 \times 2} \\
   {}& = &{20}
   \end{array}\]
   ­

Se $f’$ e $g’$ existem e se $f \circ g$ está definida, então $$\left( {f \circ g} \right)'(x) = f’\left( {g(x)} \right) \times g'(x)$$