What is the derivative of tan(x)?
sec^2(x)
The derivative of tan(x) is sec^2(x).
To obtain this result, we can use the quotient rule of differentiation, which says that for a function u(x) divided by v(x), the derivative is given by:
(u/v)’ = (u’v – uv’)/v^2
In this case, we have u(x) = sin(x) and v(x) = cos(x), so:
tan(x) = sin(x)/cos(x)
Using the quotient rule, we have:
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 trigonometric identity cos^2(x) + sin^2(x) = 1, we get:
tan'(x) = 1/(cos^2(x))
Finally, using the identity sec^2(x) = 1/(cos^2(x)), we obtain:
tan'(x) = sec^2(x)
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