Find dy/dx if. y=tanx

Sec²x

Step-by-step explanation:

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dy/dx = 1/cos²x    or  dy/dx = 1 + tan²x  or dy/dx == sec²x

Step-by-step explanation:

hello :

note :

1) dy/dx= cosx    if y = sinx

2)  dy/dx= - sinx    if y = cosx

3) cos²x +sin²x = 1   ...(**)

4)  tanx = sinx/cosx

5) d(f/g) = (g×df-f×dg)/g²...(*)

now :   let : y = tanx     and   f(x)= sinx      g(x) = cosx

 use (*)   :  dy/dx = (cosx(cosx) - sinx(-sinx)) / cos²x  = (cos²x +sin²x)/ cos²x

use (**) : dy/dx = 1/cos²x   or dy/dx = (1/cosx)² = sec²x

or : dy/dx =(cos²x +sin²x)/ cos²x = cos²x/cos²x + sin²x/ cos²x = 1+sin²x/ cos²x

dy/dx = 1 + (sinx/cosx)²

dy/dx = 1 + tan²x