The derivative of a function at a specific point is the instantaneous rate of change of the function at that point. It represents the slope of the tangent line to the function's graph at that point.
To find the derivative of a function, you must use the limit definition of derivative: f'(x) = limh→0 (f(x+h)-f(x))/h. This involves finding the slope of the secant line between two points on the function's graph and then taking the limit as the distance between those points approaches zero.
The derivative of tan(x) is sec2(x). This can be found by using the definition of derivative and the trigonometric identity: tan(x) = sin(x)/cos(x).
Yes, the derivative of tan(x) can also be expressed as 1 + tan2(x). This is another form of the derivative that can be useful in certain situations.
Finding the derivative of tan(x) can be useful in many applications, such as calculating the slope of a tangent line to a curve or finding the rate of change of a trigonometric function. It also helps in understanding the behavior and properties of tan(x) and related functions.