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

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f(x) = sin^2 x / cos^2 x

Let u simplify:

f(x) = (sinx/cox)^2

But we know that:

sinx/cosx = tanx

==> f(x) = tanx)^2

Now we will differetiate:

f(x) = 2 (tanx) (tanx)'

We know that:

tanx)' = sec^2 x

==> f'(x) = 2tanx* sec^2x

Now we know that: secx = 1/cosx

==> f'(x) = 2*(sinx/cosx) * 1/(cos^2x)

= 2sin/ cos^3 x

**==> f'(x) = 2sinx/ cos^3 x**

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 .

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