Question

Find the derivative of tan(x).

Answer 1

`\qquad\qquad\huge\underline{{\sf Answer}}♨`

Derivative of tan(x) is sec²x

`\qquad \sf  \dashrightarrow \: \therefore (d)/(dx) ( \tan(x)) = { \sec}^(2) (x)`

You can check the first principle method of derivation in attachment

Answer 2

`\rightarrow \sf (d)/(dx) (tan(x))`

`\rightarrow \sf (d)/(dx) ( \ (sin(x))/(cos(x)) \ )`

use the quotient rule

`\rightarrow \sf (cos(x) * (d)/(dx) (sin(x)-sin(x)*(d)/(dx)(cos(x) )/(cos(x)^2)`

`\rightarrow \sf (cos(x) * cos(x)-sin(x)*(-sin(x) ))/(cos(x)^2)`

`\rightarrow \sf (cos(x)^2+sin(x)^2)/(cos(x)^2)`

`\rightarrow \sf (1)/(cos(x)^2)`

`\rightarrow \sf sec(x)^2`

used formula's :

• cos²(x) + sin²(x) = 1

• 
  `\sf (1)/(cos^2(x) )= sec^2(x)`

• 
  `\sf (d)/(dx)` cos(x) = -sin(x)

• 
  `\sf (d)/(dx)` sin(x) = cos(x)

• tan(x) = sin(x)/cos(x)