What is the derivative of tan(x)?

The derivative of tan(x) is sec^2(x), which can be derived using basic differentiation rules from calculus. To understand why this is the case, we can recall that the tangent function can be expressed in terms of sine and cosine:

[ \tan(x) = \frac{\sin(x)}{\cos(x)}. ]

Using the quotient rule for derivatives, which states that if you have a function that is the quotient of two other functions, say ( \frac{u}{v} ), the derivative is given by:

[ \frac{du}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}, ]

we can apply it here with ( u = \sin(x) ) and ( v = \cos(x) ). The derivatives ( \frac{du}{dx} = \cos(x) ) and ( \frac{dv}{dx} = -\sin(x) ) give us the following:

[ \frac{d}{dx} \tan(x) = \frac{\cos(x) \cdot \cos(x) - \sin(x)(-\sin(x))}{\cos^2(x)}