(d/dx) sec(x)
To find the derivative of sec(x) with respect to x, we need to use the quotient rule
To find the derivative of sec(x) with respect to x, we need to use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then the derivative of f(x) with respect to x is given by
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2
In this case, the function we need to differentiate is f(x) = sec(x), which can also be expressed as f(x) = 1 / cos(x). Letting g(x) = 1 and h(x) = cos(x), we can differentiate them separately:
g'(x) = 0 (as 1 is a constant)
h'(x) = -sin(x) (by taking the derivative of cos(x), which is -sin(x))
Now, using the quotient rule formula, we can substitute the values and simplify:
f'(x) = (0 * cos(x) – 1 * (-sin(x))) / [cos(x)]^2
= sin(x) / cos^2(x)
= sin(x) * sec^2(x)
Thus, the derivative of sec(x) with respect to x is sin(x) * sec^2(x).
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