Differentiate f(x) = sin^2 x / cos^2 x without using the chain rule.

To differentiate the function sin^2x/cos^2x  without using
chain rule.

We know that  sin^2x/cos^2x =
tan^2x.

Therefore d/dx(sin^2x/cos^2x) =
d/dx{(tanx)^2}

d/dx (tanx)^2 = Lt (tan(x+h))^2  - (tanx
)^2}/h as h --> 0

d/dx(tanx)^2 = Lt{(tan(x+h)) -
tanx)(tan(x+h))+tax)}/h as h --> 0.

d/dx(tanx)^2 =
Lt {(1/h) (tan(xh) -tanx)}{ lt tan(x+h)+tanx} as h -->
0

d/dx(tanx)^2 = (secx)^2(2tanx) = 2(secx)^2
*tanx

Therefore d/dx{tanx)^2 = 2 (secx)^2 tanx
.