d/dx (tanx)=
(secx)∧2
The derivative of tan(x) can be found using the chain rule of differentiation. Recall that the chain rule states that if h(x) = f(g(x)), then:
h'(x) = f'(g(x)) * g'(x)
In this case, we can rewrite tan(x) as sin(x)/cos(x). Then, let f(u) = sin(u) and g(x) = cos(x), so that:
tan(x) = f(g(x)) = sin(cos(x))
Using the chain rule, we can find the derivative of tan(x) as:
d/dx (tan(x)) = d/dx (sin(cos(x))) = cos(cos(x)) * (-sin(x))
Therefore, the derivative of tan(x) is equal to -cos(x) / [cos(x)]^2 = -1/[cos(x)]^2.
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