The question asks to implicitly differentiate tan ⁡ ( x + y ) = x.I...

Nash Frank

Nash Frank

Answered

2022-07-16

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?

Answer & Explanation

Emilie Reeves

Emilie Reeves

Expert

2022-07-17Added 11 answers

From this you have: x + y = tan 1 x + n π 1 + y = 1 x 2 + 1 y = . . . .
Ciara Rose

Ciara Rose

Expert

2022-07-18Added 4 answers

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

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