How do you find the derivative of y = tan(x) using first principle?

We know that

tan(x) = sin(x) / cos(x)

d/dx tan(x) = d/dx sin(x) / cos(x)

Now will us the quotient rule for the above equation

=(cos(x) (d/dx) sin(x) – sin(x) (d-dx) cos(x) ) / cos²(x) { (d/dx) sin(x) = cos(x); (d/dx) cos(x) = -sin(x)}

=(cos(x)cos(x) + sin(x)sin(x) ) / cos²(x)

= cos²(x) + sin²(x) / cos²(x) { By trignometric identity cos²(x) + sin²(x)=1}

= 1 / cos²(x)

=sec²(x) { 1/cos²(x) = sec²(x) }

Answer

Derivative of tan x = sec²(x)