To find the derivative of tangent function, we use the quotient rule.
Recall that:
(d/dx)(f(x)/g(x)) = [g(x)(d/dx)(f(x)) – f(x)(d/dx)(g(x))]/[g(x)]^2
In this case, we have:
f(x) = sinx
g(x) = cosx
Therefore:
(d/dx)(tanx) = (d/dx)(sinx/cosx)
= [(cosx)(d/dx)(sinx) – (sinx)(d/dx)(cosx)]/[cosx]^2
= [cos^2(x) + sin^2(x)]/[cos^2(x)]
= 1/[cos^2(x)]
= sec^2(x)
Therefore, the derivative of tangent function is secant squared function.
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