Use implicit differentiation to determine ∂ z ∂ x in y z = l n ( x + z ) and sin ⁡ ( x y z ) = x + 2 y + 3 z.Here is my answer: y z = l n ( x + z ) y z ′ = ( 1 + z ′ ) 1 x + z z ′ = 1 y x + y z − 1 and sin ⁡ ( x y z ) = x + 2 y + 3 z y ( z + x z ′ ) cos ⁡ ( x y z ) = 1 + 3 z ′ y z cos ⁡ ( x y z ) + x y z ′ cos ⁡ ( x y z ) = 1 + 3 z ′ y z cos ⁡ ( x y z ) − 1 = z ′ ( 3 − x y cos ⁡ ( x y z ) ) z ′ = y z cos ⁡ ( x y z ) − 1 3 − x y cos ⁡ ( x y z ) is that right?

Question

Use implicit differentiation to determine             ∂      z              ∂      x       in   y  z  =  l  n  (  x  +  z  ) and   sin  ⁡  (  x  y  z  )  =  x  +  2  y  +  3  z.Here is my answer:  y  z  =  l  n  (  x  +  z  )  y      z    ′    =  (  1  +      z    ′    )      1          x      +      z            z    ′    =      1          y      x      +      y      z      −      1      and  sin  ⁡  (  x  y  z  )  =  x  +  2  y  +  3  z  y  (  z  +  x      z    ′    )  cos  ⁡  (  x  y  z  )  =  1  +  3      z    ′    y  z  cos  ⁡  (  x  y  z  )  +  x  y      z    ′    cos  ⁡  (  x  y  z  )  =  1  +  3      z    ′    y  z  cos  ⁡  (  x  y  z  )  −  1  =      z    ′    (  3  −  x  y  cos  ⁡  (  x  y  z  )  )      z    ′    =            y      z      cos      ⁡      (      x      y      z      )      −      1              3      −      x      y      cos      ⁡      (      x      y      z      )      is that right?

Besagnoe9 · Accepted Answer

Not quite. Don&#039;t forget the chain rule, and to include other instances of             ∂      y              ∂      x      .For instance, in the first, don&#039;t forget that      ∂          ∂      x        (  y  z  )  =            ∂      y              ∂      x        z  +  y            ∂      z              ∂      x      and in the second, don&#039;t forget that      ∂          ∂      x        (  x  +  2  y  +  3  z  )  =  1  +  2            ∂      y              ∂      x        +  3            ∂      z              ∂      x      Otherwise, your techniques look reasonably ok.

Use implicit differentiation to determine (del z)/(del x) in yz=ln(x+z) and sin(xyz)=x+2y+3z.

Answered question

Answer & Explanation

New Questions in Calculus 2