derivative of tan(x)
sec^2(x)
The derivative of tan(x) can be calculated using the quotient rule of differentiation. The quotient rule states that the derivative of a function f(x)/g(x) is given by:
(f'(x)*g(x) – f(x)*g'(x)) / g(x)^2
Applying this rule to tan(x), we have:
f(x) = sin(x)
g(x) = cos(x)
Therefore,
f'(x) = cos(x)
g'(x) = -sin(x)
Substituting these values into the quotient rule, we get:
[tan(x)]’ = (cos(x)*cos(x) – sin(x)*(-sin(x))) / [cos(x)]^2
Simplifying this expression, we get:
[tan(x)]’ = (cos^2(x) + sin^2(x)) / [cos^2(x)]
Using the identity sin^2(x) + cos^2(x) = 1, we have:
[tan(x)]’ = 1 / [cos^2(x)]
Therefore, the derivative of tan(x) is:
[tan(x)]’ = sec^2(x)
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