How to Find the Derivative of Sec(x) and Simplify it to Tan(x)

derv sec(x)

To find the derivative of sec(x), we can use the quotient rule

To find the derivative of sec(x), we can use the quotient rule. However, it’s important to note that sec(x) can also be written as 1/cos(x).

Using the quotient rule, which states that if we have a function f(x) = g(x)/h(x), the derivative is given by:

f'(x) = [g'(x)h(x) – g(x)h'(x)] / [h(x)]^2

In this case, g(x) = 1 and h(x) = cos(x). Let’s differentiate.

g'(x) = 0 (since the derivative of a constant is zero)

h'(x) = -sin(x) (the derivative of cos(x) is -sin(x), as per the chain rule)

Now, plugging these values into the quotient rule formula, we have:

f'(x) = [(0)(cos(x)) – (1)(-sin(x))] / [cos(x)]^2

Simplifying further:

f'(x) = sin(x) / cos^2(x)

Since sin(x)/cos(x) is equivalent to tan(x), the final result can be written as:

f'(x) = tan(x)

Therefore, the derivative of sec(x) is equal to tan(x).

More Answers: