Derivative of trigonometric functions

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  1. ddx  sin⁡(        )=cos⁡(        )⋅ddx(        )\frac{{d}}{{{d}x}}\;\sin \left( {\;\;\;\;} \right) = \cos \left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
    ddx  cos⁡(        )=−sin⁡(        )⋅ddx(        )\frac{{d}}{{{d}x}}\;\cos \left( {\;\;\;\;} \right) = - \sin \left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
    ddx  tan⁡(        )=sec⁡2(        )⋅ddx(        )\frac{{d}}{{{d}x}}\;\tan \left( {\;\;\;\;} \right) = {\sec ^2}\left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
    ddx  cot(        )=−csc⁡2(        )⋅ddx(        )\frac{{d}}{{{d}x}}{\;cot}\left( {\;\;\;\;} \right) = - {\csc ^2}\left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
    ddx  sec⁡(        )=sec⁡(        )tan⁡(        )⋅ddx(        )\frac{{d}}{{{d}x}}\;\sec \left( {\;\;\;\;} \right) = \sec \left( {\;\;\;\;} \right)\tan \left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
    ddx  csc⁡(        )=−csc⁡(        )cot⁡(        )⋅ddx(        )\frac{{d}}{{{d}x}}\;\csc \left( {\;\;\;\;} \right) = - \csc \left( {\;\;\;\;} \right)\cot \left( {\;\;\;\;} \right) \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
  2. Differentiate:
    a) y=sin⁡4xy = {\sin ^4}x
    b) y=sin(x4)y = sin\left( {{x^4}} \right)
  3. ddx  sin⁡(cos⁡(tan⁡x))\frac{{d}}{{{d}x}}\;\sin (\cos (\tan x))