The Chain Rule | Finding the Derivative of sec²(x) using the Quotient and Chain Rule

sec²(x) is the derivative of?

To find the derivative of sec²(x), we can use the chain rule

To find the derivative of sec²(x), we can use the chain rule.

Let’s start with the function f(x) = sec(x).

The derivative of sec(x) can be found using the quotient rule, which states that if we have a function g(x) = u(x)/v(x), where u and v are both differentiable functions, then the derivative is given by:

g'(x) = (u'(x)v(x) – u(x)v'(x)) / (v²(x))

For sec(x), we can write it as 1/cos(x), so u(x) = 1 and v(x) = cos(x).

Now let’s find the derivatives of u(x) and v(x):

u'(x) = 0 (the derivative of a constant is 0)
v'(x) = -sin(x) (the derivative of cos(x) is -sin(x))

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

sec'(x) = (0 * cos(x) – 1 * (-sin(x))) / (cos²(x))
= sin(x) / cos²(x)

Next, we need to find the derivative of sec²(x). To do this, we can use the chain rule. Recall that the chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).

In this case, our composite function is f(g(x)) = sec²(x), where g(x) = sec(x).

So, the derivative of sec²(x) is given by:

(sec²(x))’ = 2 * sec(x) * sec'(x)

Substituting the value of sec'(x) we found earlier, we have:

(sec²(x))’ = 2 * sec(x) * (sin(x) / cos²(x))

Simplifying this expression further, we get:

(sec²(x))’ = 2 * sin(x) / cos(x)

Therefore, the derivative of sec²(x) is 2 * sin(x) / cos(x).

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