The question asks to implicitly differentiate tan ⁡ ( x + y ) = x.I get the correct answer of − sin 2 ⁡ ( x + y ). However, the book also says "or you can use − x 2 x 2 + 1 ". I don't see how I can construct a proper right triangle to give/obtain this equivalent answer. Can you assist?

Question

The question asks to implicitly differentiate   tan  ⁡  (  x  +  y  )  =  x.I get the correct answer of   −      sin    2    ⁡  (  x  +  y  ). However, the book also says &quot;or you can use   −                    x        2                              x          2                +        1            &quot;. I don&#039;t see how I can construct a proper right triangle to give/obtain this equivalent answer. Can you assist?

Emilie Reeves · Accepted Answer

From this you have:   x  +  y  =      tan          −      1        ⁡  x  +  n  π    ⟹    1  +      y    ′    =            1                        x          2                +        1                ⟹        y    ′    =  .  .  .  .

Ciara Rose · Accepted Answer

Note that from the implicit equation   tan  ⁡  (  x  +  y  )  =  x you obtain   y  =  arctan  ⁡  (  x  )  −  x and therefore  −      sin    2    ⁡  (  x  +  y  )  =  −      sin    2    ⁡  (  x  +  arctan  ⁡  (  x  )  −  x  )  =  −      sin    2    ⁡  (  arctan  ⁡  (  x  )  )Now by drawing a right triangle with hypotenuse       1    +          x      2       and opposite side   x and adjacent side 1 note that  sin  ⁡  (  arctan  ⁡  (  x  )  )  =            ±      x              1      +              x        2            and thus  −      sin    2    ⁡  (  arctan  ⁡  (  x  )  )  =  −            (                        ±          x                          1          +                      x            2                              )        2    =            −              x        2                            x        2            +      1