Derivative of a function and an integral d d x ( x 6 ( ∫ 0 s i n x t d t ) )

Question

Derivative of a function and an integral

\frac{d}{d x} (x^{6} (\int_{0}^{s i n x} \sqrt{t} d t))

Colin Moran · Accepted Answer

Generally:      d          d      x            (          x      6        (          ∫              0                    s        i        n        x              f    (    t    )    d    t    )    )    =  6      x    5        ∫          0              s      i      n      x        f  (  t  )  d  t  +      x    6        d          d      x            ∫          0              s      i      n      x        f  (  t  )  d  tif we assume   F  (  x  )  =      ∫          0              x        f  (  t  )  d  t we have      d          d      x            ∫          0              s      i      n      x        f  (  t  )  d  t  =            d              d        x              F  (  sin  ⁡  x  )  =  cos  ⁡  x  f  (  sin  ⁡  x  )therefore      d          d      x            (          x      6        (          ∫              0                    s        i        n        x              f    (    t    )    d    t    )    )    =  6      x    5        ∫          0              s      i      n      x        f  (  t  )  d  t  +      x    6    cos  ⁡  x  f  (  sin  ⁡  x  )here we have      d          d      x            (          x      6        (          ∫              0                    s        i        n        x                    t        d    t    )    )    =  4      x    5    sin  ⁡  x      sin    ⁡    x    +      x    6    cos  ⁡  x      sin    ⁡    x

Derivative of a function and an integral d d x </mrow> </mfra

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