The derivative of sec(x) is sec(x)tan(x).
To understand why, let’s break it down. The secant function, sec(x), is defined as the reciprocal of the cosine function:
sec(x) = 1/cos(x)
Now, when we want to find the derivative of sec(x), we can use the quotient rule, which states that if you have a function that is the ratio of two functions, its derivative is:
(f/g)' = (f'g - fg')/g²
In our case, let’s set f = 1 and g = cos(x). The derivatives of these functions are f' = 0 and g' = -sin(x).
Applying the quotient rule, we have:
sec'(x) = (0 * cos(x) - 1 * (-sin(x))) / (cos(x))²
This simplifies to:
sec'(x) = sin(x) / cos²(x)
Since tan(x) = sin(x) / cos(x) and sec(x) = 1 / cos(x), we can rewrite the derivative as:
sec'(x) = sec(x)tan(x)
So, in conclusion, the derivative of sec(x) is sec(x)tan(x).