I am struggling with the following problem for implicit differentiation.I am tasked to differentiate implicitly the following function, and evaluate y ″ ( 0 ), where y = y ( x ). ln ⁡ ( y + 1 ) + sin ⁡ ( x y ) = ln ⁡ ( 5 ) .I have differentiated this once to find, ( y + x y ′ ) cos ⁡ ( x y ) + y ′ y + 1 = 0But how to advance from here to find y ″ ( 0 )?

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I am struggling with the following problem for implicit differentiation.I am tasked to differentiate implicitly the following function, and evaluate       y    ″    (  0  ), where   y  =  y  (  x  ).  ln  ⁡  (  y  +  1  )  +  sin  ⁡  (  x  y  )  =  ln  ⁡  (  5  )  .I have differentiated this once to find,  (  y  +  x      y    ′    )  cos  ⁡  (  x  y  )  +            y      ′              y      +      1        =  0But how to advance from here to find       y    ″    (  0  )?

Donovan Shields · Accepted Answer

In the original equation we can find   y  (  0  ):  ln  ⁡  (  y  +  1  )  +  sin  ⁡  0  =  ln  ⁡  5    ⟹    y  (  0  )  =  4And in that equation we can find       y    ′    (  0  ):  (  4  +  0  )  cos  ⁡  0  +            y      ′        5    =  0    ⟹        y    ′    (  0  )  =  −  20Now just implicit differentiate that expression again and follow the same procedure. Can you take it from here?

I am struggling with the following problem for implicit differentiation. I am tasked to differentiate implicitly the following function, and evaluate y′′(0), where y=y(x). ln(y+1)+sin(xy)=ln(5).

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