A Comprehensive Guide: Finding the Derivative of Sec(x) Using Quotient Rule and Chain Rule

Derivative of sec(x)

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

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

Option 1: Using the Quotient Rule
The quotient rule states that for two functions f(x) and g(x), the derivative of their ratio h(x) = f(x) / g(x) is given by:

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

Here, f(x) = 1 and g(x) = cos(x).

We know that the derivative of cos(x) is -sin(x), so f'(x) = 0 and g'(x) = -sin(x).

Plugging these values into the quotient rule formula, we have:

h'(x) = (0 * cos(x) – 1 * (-sin(x))) / (cos(x))^2
= sin(x) / (cos(x))^2
= sin(x) / cos^2(x)
= tan(x) / cos(x)

Therefore, the derivative of sec(x) using the quotient rule is tan(x) / cos(x).

Option 2: Using the Chain Rule
Another way to find the derivative of sec(x) is by using the chain rule. The chain rule states that if y = f(g(x)), then the derivative dy/dx is given by:

dy/dx = f'(g(x)) * g'(x)

In this case, f(u) = sec(u) and g(x) = x.

We know that the derivative of sec(u) is sec(u) * tan(u), so f'(u) = sec(u) * tan(u).

Using the chain rule, we have:

dy/dx = f'(g(x)) * g'(x)
= sec(g(x)) * tan(g(x)) * 1
= sec(x) * tan(x)

Therefore, the derivative of sec(x) using the chain rule is sec(x) * tan(x).

Both methods yield the same result, so we conclude that the derivative of sec(x) is equal to tan(x) / cos(x) or sec(x) * tan(x).

Please note that the domain of sec(x) excludes the values where cos(x) = 0, as the function is undefined there.

More Answers: