a)Apply the chain rule to u = xyz, where x, y, and z are functions of t, to get a rule for differentiating a product of three functions of one variable.b)Derive the rule in (a) by using one-variable calculus.

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beechenUSR · Accepted Answer

u  =  x  y  z           x  =  f  (  t  )        y  =  g  (  t  )        z  =  h  (  t  )              d      u              d      t        =            ∂      u              ∂      x        ⋅            d      x              d      t        +            ∂      u              ∂      y        ⋅            d      y              d      t        +            ∂      u              ∂      z        ⋅            d      z              d      t          u  =  x  y  z              ∂      u              ∂      x        =      ∂          ∂      x        (  x  y  z  )  =  y  z      ∂          ∂      x        (  x  )  =  y  z  (  1  )  =  y  z              ∂      u              ∂      y        =      ∂          ∂      y        (  x  y  z  )  =  x  z  ⋅      ∂          ∂      y        (  y  )  =  x  z  (  1  )  =  x  z              ∂      u              ∂      z        =      ∂          ∂      z        (  x  y  z  )  =  x  y      ∂          ∂      z        (  z  )  =  x  y  (  1  )  =  x  y              d      u              d      t        =  y  z  ⋅            d      x              d      t        +  x  z  ⋅            d      y              d      t        +  x  y  ⋅            d      z              d      t

Apply the chain rule to u = xyz, where x, y, and z are functions of t, to get a rule for differentiating a product of three functions of one variable.

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