Derivative of Tan
d/dx tan(x) = sec²(x)
The derivative of tan(x) is given by:
(d/dx) tan(x) = sec^2(x)
where sec(x) is the secant function, defined as 1/cos(x).
The proof of this formula comes from the quotient rule of differentiation, which states that the derivative of the quotient of two functions is given by:
(d/dx) (f(x)/g(x)) = [g(x) * f'(x) – f(x) * g'(x)]/[g(x)]^2
Applying this rule to the function f(x) = sin(x) and g(x) = cos(x), we get:
(d/dx) tan(x) = (cos^2(x) * sin'(x) – sin(x) * cos(x) * cos'(x))/[cos^2(x)]^2
= (cos^2(x) * cos(x) – sin^2(x))/[cos^4(x)]
= cos(x)/cos^2(x)
= 1/cos(x)^2
= sec^2(x)
Therefore, the derivative of tan(x) is equal to sec^2(x).
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